Combinatorics Wiki - User contributions [en-gb]

TempArcTransitive

2024-10-01T10:32:01Z

Roblewis:

'''NB This page is incomplete and still under construction.'''

===Table of the orders of the largest known arc-transitive graphs for the undirected degree diameter problem===

Below is the table of the largest known arc-transitive graphs in the undirected [[The Degree/Diameter Problem | degree diameter problem]] for graphs of [http://en.wikipedia.org/wiki/Degree_(graph_theory) degree] 3 ≤ ''d'' ≤ 20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2 ≤ ''k'' ≤ 10. All graphs which are known to be optimal are marked in '''bold'''.

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10'''
|-
| '''3''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''4''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''5''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''6''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''7''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''8''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''9''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''10''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''11''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''12''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''13''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''14''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''15''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''16''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''17''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''18''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''19''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''20''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #eee; text-align: center;" | * || Details.
|-
|}
</center>

===Table of the orders of the largest known arc-transitive circulant graphs for the undirected degree diameter problem===

Below is the table of the largest known arc-transitive circulant graphs in the undirected [[The Degree/Diameter Problem | degree diameter problem]] for graphs of [http://en.wikipedia.org/wiki/Degree_(graph_theory) degree] 3 ≤ ''d'' ≤ 20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2 ≤ ''k'' ≤ 10. All graphs which are known to be optimal are marked in '''bold'''.

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10'''
|-
| '''3''' || style="background-color: #eee;" | 6 || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''4''' || style="background-color: #eee;" | 13 || style="background-color: #eee;" | 25 || style="background-color: #eee;" | 41 ||style="background-color: #eee;" | 61 || style="background-color: #eee;" | 85 ||style="background-color: #eee;" | 113 ||style="background-color: #eee;" | 145 ||style="background-color: #eee;" | 181 ||style="background-color: #eee;" | 221
|-
| '''5''' || style="background-color: #eee;" | 10 || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''6''' || style="background-color: #eee;" | 19 || style="background-color: #eee;" | 38 || style="background-color: #eee;" | 117 ||style="background-color: #eee;" | 171 || style="background-color: #eee;" | 279 ||style="background-color: #eee;" | 515 ||style="background-color: #eee;" | 695 ||style="background-color: #eee;" | 905 ||style="background-color: #eee;" | 1393
|-
| '''7''' || style="background-color: #eee;" | 14 || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''8''' || style="background-color: #eee;" | 17 || style="background-color: #eee;" | 75 || style="background-color: #eee;" | 146 ||style="background-color: #eee;" | 401 || style="background-color: #eee;" | 777 ||style="background-color: #eee;" | 1365 ||style="background-color: #eee;" | 2129 ||style="background-color: #eee;" | 3281 ||style="background-color: #eee;" | 5081
|-
| '''9''' || style="background-color: #eee;" | 18 || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''10''' || style="background-color: #eee;" | 41 || style="background-color: #eee;" | 151 || style="background-color: #eee;" | 363 ||style="background-color: #eee;" | 1023 || style="background-color: #eee;" | 1623 ||style="background-color: #eee;" | 4443 ||style="background-color: #eee;" | 6663 ||style="background-color: #eee;" | 11163 ||style="background-color: #eee;" | 18483
|-
| '''11''' || style="background-color: #eee;" | 22 || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''12''' || style="background-color: #eee;" | 57 || style="background-color: #eee;" | 185 || style="background-color: #eee;" | 785 ||style="background-color: #eee;" | 1685 || style="background-color: #eee;" | 3965 ||style="background-color: #eee;" | 10777 ||style="background-color: #eee;" | 15769 ||style="background-color: #eee;" | 35269 ||style="background-color: #eee;" | 52429
|-
| '''13''' || style="background-color: #eee;" | 26 || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''14''' || style="background-color: #eee;" | 71 || style="background-color: #eee;" | 379 || style="background-color: #eee;" | 953 ||style="background-color: #eee;" | 3067 || style="background-color: #eee;" | 8023 ||style="background-color: #eee;" | 18415 ||style="background-color: #eee;" | 37803 ||style="background-color: #eee;" | 72327 ||style="background-color: #eee;" | 196689
|-
| '''15''' || style="background-color: #eee;" | 30 || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''16''' || style="background-color: #eee;" | 97 || style="background-color: #eee;" | 401 || style="background-color: #eee;" | 1649 ||style="background-color: #eee;" | 4947 || style="background-color: #eee;" | 14081 ||style="background-color: #eee;" | 35921 ||style="background-color: #eee;" | 80257 ||style="background-color: #eee;" | 173969 ||style="background-color: #eee;" | 354433
|-
| '''17''' || style="background-color: #eee;" | 34 || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''18''' || style="background-color: #eee;" | ? || style="background-color: #eee;" | ? || style="background-color: #eee;" | 1393 ||style="background-color: #eee;" | ? || style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ?
|-
| '''19''' || style="background-color: #eee;" | 38 || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''20''' || style="background-color: #eee;" | 101 || style="background-color: #eee;" | ? || style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? || style="background-color: #eee;" | ? ||style="background-color: #eee;" | 106025 ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ?
|-
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #eee; text-align: center;" | * || Details.
|-
|}
</center>

TempArcTransitive

2024-09-30T19:36:46Z

Roblewis:

'''NB This page is incomplete and still under construction.'''

===Table of the orders of the largest known arc-transitive graphs for the undirected degree diameter problem===

Below is the table of the largest known arc-transitive graphs in the undirected [[The Degree/Diameter Problem | degree diameter problem]] for graphs of [http://en.wikipedia.org/wiki/Degree_(graph_theory) degree] 3 ≤ ''d'' ≤ 20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2 ≤ ''k'' ≤ 10. All graphs which are known to be optimal are marked in '''bold'''.

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10'''
|-
| '''3''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''4''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''5''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''6''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''7''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''8''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''9''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''10''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''11''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''12''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''13''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''14''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''15''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''16''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''17''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''18''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''19''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''20''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #eee; text-align: center;" | * || Details.
|-
|}
</center>

===Table of the orders of the largest known arc-transitive circulant graphs for the undirected degree diameter problem===

Below is the table of the largest known arc-transitive circulant graphs in the undirected [[The Degree/Diameter Problem | degree diameter problem]] for graphs of [http://en.wikipedia.org/wiki/Degree_(graph_theory) degree] 3 ≤ ''d'' ≤ 20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2 ≤ ''k'' ≤ 10. All graphs which are known to be optimal are marked in '''bold'''.

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10'''
|-
| '''3''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''4''' || style="background-color: #eee;" | 13 || style="background-color: #eee;" | 25 || style="background-color: #eee;" | 41 ||style="background-color: #eee;" | 61 || style="background-color: #eee;" | 85 ||style="background-color: #eee;" | 113 ||style="background-color: #eee;" | 145 ||style="background-color: #eee;" | 181 ||style="background-color: #eee;" | 221
|-
| '''5''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''6''' || style="background-color: #eee;" | 19 || style="background-color: #eee;" | 38 || style="background-color: #eee;" | 117 ||style="background-color: #eee;" | 171 || style="background-color: #eee;" | 279 ||style="background-color: #eee;" | 515 ||style="background-color: #eee;" | 695 ||style="background-color: #eee;" | 905 ||style="background-color: #eee;" | 1393
|-
| '''7''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''8''' || style="background-color: #eee;" | 17 || style="background-color: #eee;" | 75 || style="background-color: #eee;" | 146 ||style="background-color: #eee;" | 401 || style="background-color: #eee;" | 777 ||style="background-color: #eee;" | 1365 ||style="background-color: #eee;" | 2129 ||style="background-color: #eee;" | 3281 ||style="background-color: #eee;" | 5081
|-
| '''9''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''10''' || style="background-color: #eee;" | 41 || style="background-color: #eee;" | 151 || style="background-color: #eee;" | 363 ||style="background-color: #eee;" | 1023 || style="background-color: #eee;" | 1623 ||style="background-color: #eee;" | 4443 ||style="background-color: #eee;" | 6663 ||style="background-color: #eee;" | 11163 ||style="background-color: #eee;" | 18483
|-
| '''11''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''12''' || style="background-color: #eee;" | 57 || style="background-color: #eee;" | 185 || style="background-color: #eee;" | 785 ||style="background-color: #eee;" | 1685 || style="background-color: #eee;" | 3965 ||style="background-color: #eee;" | 10777 ||style="background-color: #eee;" | 15769 ||style="background-color: #eee;" | 35269 ||style="background-color: #eee;" | 52429
|-
| '''13''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''14''' || style="background-color: #eee;" | 71 || style="background-color: #eee;" | 379 || style="background-color: #eee;" | 953 ||style="background-color: #eee;" | 3067 || style="background-color: #eee;" | 8023 ||style="background-color: #eee;" | 18415 ||style="background-color: #eee;" | 37803 ||style="background-color: #eee;" | 72327 ||style="background-color: #eee;" | 196689
|-
| '''15''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''16''' || style="background-color: #eee;" | 97 || style="background-color: #eee;" | 401 || style="background-color: #eee;" | 1649 ||style="background-color: #eee;" | 4947 || style="background-color: #eee;" | 14081 ||style="background-color: #eee;" | 35921 ||style="background-color: #eee;" | 80257 ||style="background-color: #eee;" | 173969 ||style="background-color: #eee;" | 354433
|-
| '''17''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''18''' || style="background-color: #eee;" | ? || style="background-color: #eee;" | ? || style="background-color: #eee;" | 1393 ||style="background-color: #eee;" | ? || style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ?
|-
| '''19''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''20''' || style="background-color: #eee;" | 101 || style="background-color: #eee;" | ? || style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? || style="background-color: #eee;" | ? ||style="background-color: #eee;" | 106025 ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ?
|-
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #eee; text-align: center;" | * || Details.
|-
|}
</center>

TempArcTransitive

2024-09-30T18:53:01Z

Roblewis:

'''NB This page is incomplete and still under construction.'''

===Table of the orders of the largest known arc-transitive graphs for the undirected degree diameter problem===

Below is the table of the largest known arc-transitive graphs in the undirected [[The Degree/Diameter Problem | degree diameter problem]] for graphs of [http://en.wikipedia.org/wiki/Degree_(graph_theory) degree] 3 ≤ ''d'' ≤ 20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2 ≤ ''k'' ≤ 10. All graphs which are known to be optimal are marked in '''bold'''.

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10'''
|-
| '''3''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''4''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''5''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''6''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''7''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''8''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''9''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''10''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''11''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''12''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''13''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''14''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''15''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''16''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''17''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''18''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''19''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
| '''20''' || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 || style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999 ||style="background-color: #eee;" | 999
|-
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #eee; text-align: center;" | * || Details.
|-
|}
</center>

===Table of the orders of the largest known arc-transitive circulant graphs for the undirected degree diameter problem===

Below is the table of the largest known arc-transitive circulant graphs in the undirected [[The Degree/Diameter Problem | degree diameter problem]] for graphs of [http://en.wikipedia.org/wiki/Degree_(graph_theory) degree] 3 ≤ ''d'' ≤ 20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2 ≤ ''k'' ≤ 10. All graphs which are known to be optimal are marked in '''bold'''.

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10'''
|-
| '''3''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''4''' || style="background-color: #eee;" | 13 || style="background-color: #eee;" | 25 || style="background-color: #eee;" | 41 ||style="background-color: #eee;" | 61 || style="background-color: #eee;" | 85 ||style="background-color: #eee;" | 113 ||style="background-color: #eee;" | 145 ||style="background-color: #eee;" | 181 ||style="background-color: #eee;" | 221
|-
| '''5''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''6''' || style="background-color: #eee;" | 19 || style="background-color: #eee;" | 38 || style="background-color: #eee;" | 117 ||style="background-color: #eee;" | 171 || style="background-color: #eee;" | 279 ||style="background-color: #eee;" | 515 ||style="background-color: #eee;" | 695 ||style="background-color: #eee;" | 905 ||style="background-color: #eee;" | 1393
|-
| '''7''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''8''' || style="background-color: #eee;" | 17 || style="background-color: #eee;" | 75 || style="background-color: #eee;" | 146 ||style="background-color: #eee;" | 401 || style="background-color: #eee;" | 777 ||style="background-color: #eee;" | 1365 ||style="background-color: #eee;" | 2129 ||style="background-color: #eee;" | 3281 ||style="background-color: #eee;" | 5081
|-
| '''9''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''10''' || style="background-color: #eee;" | 41 || style="background-color: #eee;" | 151 || style="background-color: #eee;" | 363 ||style="background-color: #eee;" | 1023 || style="background-color: #eee;" | 1623 ||style="background-color: #eee;" | 4443 ||style="background-color: #eee;" | 6663 ||style="background-color: #eee;" | 11163 ||style="background-color: #eee;" | 18483
|-
| '''11''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''12''' || style="background-color: #eee;" | 57 || style="background-color: #eee;" | 185 || style="background-color: #eee;" | 785 ||style="background-color: #eee;" | 1685 || style="background-color: #eee;" | 3965 ||style="background-color: #eee;" | 10777 ||style="background-color: #eee;" | 1769? ||style="background-color: #eee;" | 35269 ||style="background-color: #eee;" | 52429
|-
| '''13''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''14''' || style="background-color: #eee;" | 71 || style="background-color: #eee;" | 379 || style="background-color: #eee;" | 953 ||style="background-color: #eee;" | 3067 || style="background-color: #eee;" | 8023 ||style="background-color: #eee;" | 18415 ||style="background-color: #eee;" | 37803 ||style="background-color: #eee;" | 72327 ||style="background-color: #eee;" | 196689
|-
| '''15''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''16''' || style="background-color: #eee;" | 97 || style="background-color: #eee;" | 401 || style="background-color: #eee;" | 1649 ||style="background-color: #eee;" | 4947 || style="background-color: #eee;" | 14081 ||style="background-color: #eee;" | 35921 ||style="background-color: #eee;" | 80257 ||style="background-color: #eee;" | 173969 ||style="background-color: #eee;" | 354433
|-
| '''17''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''18''' || style="background-color: #eee;" | ? || style="background-color: #eee;" | ? || style="background-color: #eee;" | 1393 ||style="background-color: #eee;" | ? || style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ?
|-
| '''19''' || style="background-color: #eee;" | - || style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - || style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | - ||style="background-color: #eee;" | -
|-
| '''20''' || style="background-color: #eee;" | 101 || style="background-color: #eee;" | ? || style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? || style="background-color: #eee;" | ? ||style="background-color: #eee;" | 106025 ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ? ||style="background-color: #eee;" | ?
|-
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #eee; text-align: center;" | * || Details.
|-
|}
</center>

The Degree Diameter Problem for Circulant Graphs

2019-11-24T09:24:53Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: magenta;" | '''16''' ||style="background-color: magenta;" | '''36''' ||style="background-color: magenta;" | '''64''' ||style="background-color: magenta;" | '''100''' ||style="background-color: magenta;" | '''144''' ||style="background-color: magenta;" | '''196''' ||style="background-color: magenta;" | '''256''' ||style="background-color: magenta;" | '''324''' ||style="background-color: magenta;" | '''400''' ||style="background-color: magenta;" | '''484''' ||style="background-color: magenta;" | '''576''' ||style="background-color: magenta;" | '''676''' ||style="background-color: magenta;" | '''784''' ||style="background-color: magenta;" | '''900''' ||style="background-color: magenta;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 45 900 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 425 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 208 872 ||style="background-color: #66ff66;" | 492 776 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 31 440 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 258 040 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 416 256 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 328 220 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 42 252 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 371 400 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 947 880 ||style="background-color: #66ff66;" | 10 238 745 ||style="background-color: #66ff66;" | 20 452 920 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 126 967 008
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|24%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-10-14T21:35:31Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 45 900 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 425 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 208 872 ||style="background-color: #66ff66;" | 492 776 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 31 440 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 258 040 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 416 256 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 328 220 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 42 252 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 371 400 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 947 880 ||style="background-color: #66ff66;" | 10 238 745 ||style="background-color: #66ff66;" | 20 452 920 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 126 967 008
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|24%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-10-14T16:15:44Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 425 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 208 872 ||style="background-color: #66ff66;" | 492 776 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 31 440 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 258 040 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 416 256 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 328 220 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 42 252 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 371 400 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 947 880 ||style="background-color: #66ff66;" | 10 238 745 ||style="background-color: #66ff66;" | 20 452 920 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 126 967 008
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|24%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-10-07T16:20:58Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 425 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 208 872 ||style="background-color: #66ff66;" | 492 776 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 258 040 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 416 256 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 42 252 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 371 400 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 947 880 ||style="background-color: #66ff66;" | 10 238 745 ||style="background-color: #66ff66;" | 20 452 920 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 126 967 008
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|24%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-09-11T13:45:21Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 208 872 ||style="background-color: #66ff66;" | 492 776 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 42 252 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 371 400 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 947 880 ||style="background-color: #66ff66;" | 10 238 745 ||style="background-color: #66ff66;" | 20 452 920 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 126 967 008
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|24%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-09-11T13:44:38Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 208 872 ||style="background-color: #66ff66;" | 492 776 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 42 252 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 371 400 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 947 880 ||style="background-color: #66ff66;" | 10 238 745 ||style="background-color: #66ff66;" | 20 452 920 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 126 967 008
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|23%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-09-10T15:29:17Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 208 872 ||style="background-color: #66ff66;" | 492 776 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 42 252 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 371 400 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 901 304 ||style="background-color: #66ff66;" | 10 238 745 ||style="background-color: #66ff66;" | 20 221 420 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 125 721 300
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|23%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-09-03T21:30:26Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 208 872 ||style="background-color: #66ff66;" | 492 776 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 238 745 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|22%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-09-03T21:27:58Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 208 772 ||style="background-color: #66ff66;" | 492 776 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 238 745 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|22%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-25T23:10:28Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 238 745 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|22%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-20T20:26:09Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|22%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-20T20:25:11Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 732 560
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 140 800
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 785 072
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 15 158 272
|-
|41%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 405 905
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 546 705
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 24 331 777
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 490 049
|-
|34%
|}

|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}

| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 079 250
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 952 610
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 38 878 482
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 63 821 826
|-
|33%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 298 155
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 50 250 765
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 89 129 247
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 152 951 073
|-
|28%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 41 517 060
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 77 548 920
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 139 380 012
|-
|27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 242 080 320
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 81 270 333
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 158 819 253
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 298 199 265
|-
|24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 540 279 585
|-
|22%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-20T19:50:46Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-20T19:49:44Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 820
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301 560
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|423 092
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|581 184
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|579 125
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|880 685
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 303 777
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 884 961
|-
|51%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596 610
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|948 430
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 459 810
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 184 462
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 188 738
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 340 495
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 800 305
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 984 767
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 173 505
|-
|42%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-20T15:17:05Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|52
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|60
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|365
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|421
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|481
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|545
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|676
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|784
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|900
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1024
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 943
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|3 629
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|4 431
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|5 357
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 928
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 392
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 080
|-
|89%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 008
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 569
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|29 961
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 041
|-
|75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|50 049
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39 210
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52 530
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|69 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|89 090
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|124 515
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|177 045
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|246 047
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|335 137
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-20T10:07:48Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: #66ff66;" | 128 852 ||style="background-color: #66ff66;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-20T10:07:18Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: #66ff66;" | 77 572 ||style="background-color: #66ff66;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: orange;" | 128 852 ||style="background-color: orange;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-20T10:06:32Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12''' || '''13''' || '''14''' || '''15''' || '''16'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48''' ||style="background-color: #bbffff;" | '''52'''||style="background-color: #bbffff;" | '''56''' ||style="background-color: #bbffff;" | '''60'''||style="background-color: #bbffff;" | '''64'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313''' ||style="background-color: beige;" | '''365''' ||style="background-color: beige;" | '''421''' ||style="background-color: beige;" | '''481''' ||style="background-color: beige;" | '''545'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576''' ||style="background-color: #66cc66;" | '''676''' ||style="background-color: #66cc66;" | '''784''' ||style="background-color: #66cc66;" | '''900''' ||style="background-color: #66cc66;" | '''1 024'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329''' ||style="background-color: magenta;" | '''2 943''' ||style="background-color: magenta;" | '''3 629''' ||style="background-color: magenta;" | '''4 431''' ||style="background-color: magenta;" | '''5 357'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144 ||style="background-color: magenta;" | 5 236 ||style="background-color: magenta;" | 6 536 ||style="background-color: magenta;" | 8 060 ||style="background-color: magenta;" | 9 744
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552 ||style="background-color: #66ff66;" | 17 024 ||style="background-color: #66ff66;" | 22 568 ||style="background-color: #66ff66;" | 29 408 ||style="background-color: #66ff66;" | 37 664
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192 ||style="background-color: #66ff66;" | 29 068 ||style="background-color: #66ff66;" | 39 032 ||style="background-color: #66ff66;" | 51 300 ||style="background-color: #66ff66;" | 66 336
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025 ||style="background-color: orange;" | 77 572 ||style="background-color: orange;" | 110 045 ||style="background-color: #66ff66;" | 152 671 ||style="background-color: #66ff66;" | 208 052
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248 || style="background-color: orange;" | 128 852 ||style="background-color: orange;" | 184 424 ||style="background-color: #66ff66;" | 259 260 ||style="background-color: #66ff66;" | 355 576
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 448 920 ||style="background-color: #66ff66;" | 662 680 ||style="background-color: #66ff66;" | 952 985
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840 ||style="background-color: #66ff66;" | 476 100 ||style="background-color: #66ff66;" | 732 744 ||style="background-color: #66ff66;" | 1 081 860 ||style="background-color: #66ff66;" | 1 593 064
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989 || style="background-color: #66ff66;" | 996 240 ||style="background-color: #66ff66;" | 1 603 216 ||style="background-color: #66ff66;" | 2 486 227 || style="background-color: #66ff66;" | 3 843 540
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480 ||style="background-color: #66ff66;" | 1 550 228 ||style="background-color: #66ff66;" | 2 566 712 ||style="background-color: #66ff66;" | 4 013 468 ||style="background-color: #66ff66;" | 6 155 056
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428 ||style="background-color: #66ff66;" | 2 982 623 ||style="background-color: #66ff66;" | 5 209 347 ||style="background-color: #66ff66;" | 8 476 048 ||style="background-color: #66ff66;" | 13 588 848
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104 ||style="background-color: #66ff66;" | 4 557 364 ||style="background-color: #66ff66;" | 7 729 000 ||style="background-color: #66ff66;" | 13 275 108 ||style="background-color: #66ff66;" | 21 252 864
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640 ||style="background-color: #66ff66;" | 8 068 383 ||style="background-color: #66ff66;" | 14 718 984 ||style="background-color: #66ff66;" | 25 609 955 ||style="background-color: #66ff66;" | 43 068 508
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520 ||style="background-color: #66ff66;" | 11 797 684 ||style="background-color: #66ff66;" | 21 659 528 ||style="background-color: #66ff66;" | 38 156 524 ||style="background-color: #66ff66;" | 66 601 304
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820 ||style="background-color: #66ff66;" | 19 505 285 ||style="background-color: #66ff66;" | 38 155 632 ||style="background-color: #66ff66;" | 70 612 644 ||style="background-color: #66ff66;" | 119 170 289
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-19T15:41:05Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 078 280 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-13T17:07:30Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 80 940 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 065 020 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-10T16:39:12Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 78 513 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 065 020 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-10T13:51:56Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 78 513 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 061 864 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-10T13:50:33Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 78 513 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 061 864 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|457
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 099
|-
|100%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|100%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-10T13:45:24Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 78 513 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 061 864 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|681
|-
|67%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 683
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|912
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|57%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|40%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-10T13:38:06Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 78 513 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 061 864 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 30 864 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 100 520
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|681
|-
|67%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 683
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|912
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|29%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-08T12:28:17Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 77 577 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 061 864 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 29 928 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 037 496
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|681
|-
|67%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 683
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|912
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-08-03T13:32:09Z

Roblewis:

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: gold;" | 655 ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 77 577 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 024 915 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: gold;" | 722 ||style="background-color: gold;" | 2 696 ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 29 928 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 037 496
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: gold;" | 815 ||style="background-color: gold;" | 3 175 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 132 720 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 121 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|681
|-
|67%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 683
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|912
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-06-07T14:45:54Z

Roblewis: /* Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs */

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: ;" | ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 77 577 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 024 915 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: ;" | ||style="background-color: ;" | ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 29 928 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 037 496
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: ;" | ||style="background-color: #66ff66;" | 2 828 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 131 835 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 001 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|681
|-
|67%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 683
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|912
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}
|-

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}
|-

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|28%
|}
|-

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-06-07T14:45:01Z

Roblewis: /* Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs */

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: ;" | ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 77 577 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 024 915 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: ;" | ||style="background-color: ;" | ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 29 928 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 037 496
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: ;" | ||style="background-color: #66ff66;" | 2 828 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 131 835 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 001 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|681
|-
|67%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 683
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|912
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|28%
|}

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-06-07T14:43:32Z

Roblewis: /* Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs */

===Table of the orders of the largest known circulant graphs===

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''8'''||style="background-color: #bbffff;" | '''12'''||style="background-color: #bbffff;" | '''16''' ||style="background-color: #bbffff;" | '''20''' ||style="background-color: #bbffff;" | '''24''' ||style="background-color: #bbffff;" | '''28'''||style="background-color: #bbffff;" | '''32''' ||style="background-color: #bbffff;" | '''36''' || style="background-color: #bbffff;" | '''40''' ||style="background-color: #bbffff;" | '''44'''||style="background-color: #bbffff;" | '''48'''
|-
| '''4''' ||style="background-color: beige;" | '''13'''|| style="background-color: beige;" | '''25''' ||style="background-color: beige;" | '''41''' || style="background-color: beige;" | '''61''' ||style="background-color: beige;" | '''85''' ||style="background-color: beige;" | '''113''' ||style="background-color: beige;" | '''145''' ||style="background-color: beige;" | '''181''' ||style="background-color: beige;" | '''221''' ||style="background-color: beige;" | '''265''' ||style="background-color: beige;" | '''313'''
|-
| '''5''' ||style="background-color: white;" | '''16''' ||style="background-color: #66cc66;" | '''36''' ||style="background-color: #66cc66;" | '''64''' ||style="background-color: #66cc66;" | '''100''' ||style="background-color: #66cc66;" | '''144''' ||style="background-color: #66cc66;" | '''196''' ||style="background-color: #66cc66;" | '''256''' ||style="background-color: #66cc66;" | '''324''' ||style="background-color: #66cc66;" | '''400''' ||style="background-color: #66cc66;" | '''484''' ||style="background-color: #66cc66;" | '''576'''
|-
| '''6''' ||style="background-color: magenta;" | '''21''' ||style="background-color: magenta;" | '''55''' ||style="background-color: magenta;" | '''117''' ||style="background-color: magenta;" | '''203''' ||style="background-color: magenta;" | '''333''' ||style="background-color: magenta;" | '''515''' ||style="background-color: magenta;" | '''737''' ||style="background-color: magenta;" | '''1 027''' ||style="background-color: magenta;" | '''1 393''' ||style="background-color: magenta;" | '''1 815''' ||style="background-color: magenta;" | '''2 329'''
|-
| '''7''' ||style="background-color: magenta;" | '''26''' ||style="background-color: magenta;" | '''76''' ||style="background-color: magenta;" | '''160''' ||style="background-color: magenta;" | '''308''' ||style="background-color: magenta;" | '''536''' ||style="background-color: magenta;" | '''828''' ||style="background-color: magenta;" | '''1 232''' ||style="background-color: magenta;" | '''1 764''' ||style="background-color: magenta;" | '''2 392''' ||style="background-color: magenta;" | 3 180 ||style="background-color: magenta;" | 4 144
|-
| '''8''' ||style="background-color: #81BEF7;" | '''35''' ||style="background-color: #CCFF00;" | '''104''' ||style="background-color: #CCFF00;" | '''248''' ||style="background-color: #CCFF00;" | '''528''' ||style="background-color: #66ff66;" | '''984''' ||style="background-color: #66ff66;" | '''1 712''' ||style="background-color: #66ff66;" | 2 768 ||style="background-color: #66ff66;" | 4 280 ||style="background-color: #66ff66;" | 6 320 ||style="background-color: #66ff66;" | 9 048 ||style="background-color: #66ff66;" | 12 552
|-
| '''9''' ||style="background-color: #81BEF7;" | '''42''' || style="background-color: #66cc66;" | '''130''' ||style="background-color: #CCFF00;" | '''320''' ||style="background-color: #66ff66;" | '''700''' ||style="background-color: #66ff66;" | '''1 416''' ||style="background-color: #66ff66;" | 2 548 ||style="background-color: #66ff66;" | 4 304 ||style="background-color: #66ff66;" | 6 804 ||style="background-color: #66ff66;" | 10 320 ||style="background-color: #66ff66;" | 15 004 ||style="background-color: #66ff66;" | 21 192
|-
| '''10''' ||style="background-color: #81BEF7;" | '''51''' || style="background-color: #66cc66;" | '''177''' ||style="background-color: yellow;" | '''457''' ||style="background-color: #66ff66;" | '''1 099''' ||style="background-color: orange;" | 2 380 ||style="background-color: orange;" | 4 551 ||style="background-color: orange;" | 8 288 ||style="background-color: orange;" | 14 099 ||style="background-color: orange;" | 22 805 ||style="background-color: #66ff66;" | 35 568 ||style="background-color: #66ff66;" | 53 025
|-
| '''11''' ||style="background-color: #81BEF7;" | '''56''' ||style="background-color: yellow;" | '''210''' || style="background-color: yellow;" | '''576''' ||style="background-color: orange;" | 1 428 ||style="background-color: orange;" | 3 200 ||style="background-color: orange;" | 6 652 ||style="background-color: orange;" | 12 416 || style="background-color: orange;" | 21 572 ||style="background-color: orange;" | 35 880 ||style="background-color: #66ff66;" | 56 700 ||style="background-color: #66ff66;" | 87 248
|-
| '''12''' ||style="background-color: #81BEF7;" | '''67''' ||style="background-color: yellow;" | '''275''' ||style="background-color: orange;" | 819 ||style="background-color: #66ff66;" | 2 120 ||style="background-color: #66ff66;" | 5 044 ||style="background-color: #66ff66;" | 10 777 ||style="background-color: #66ff66;" | 21 384 ||style="background-color: #66ff66;" | 39 996 ||style="background-color: #66ff66;" | 69 965 ||style="background-color: #66ff66;" | 117 712 ||style="background-color: #66ff66;" | 190 392
|-
| '''13''' ||style="background-color: #81BEF7;" | '''80''' ||style="background-color: yellow;" | '''312''' ||style="background-color: orange;" | 970 ||style="background-color: #66ff66;" | 2 676 ||style="background-color: #66ff66;" | 6 256 ||style="background-color: #66ff66;" | 14 740 ||style="background-color: #66ff66;" | 30 760 ||style="background-color: #66ff66;" | 57 396 ||style="background-color: #66ff66;" | 106 120 ||style="background-color: #66ff66;" | 182 980 ||style="background-color: #66ff66;" | 295 840
|-
| '''14''' ||style="background-color: #81BEF7;" | '''90''' ||style="background-color: yellow;" | '''381''' || style="background-color: orange;" | 1 229 ||style="background-color: #66ff66;" | 3 695 ||style="background-color: #66ff66;" | 9 800 ||style="background-color: #66ff66;" | 23 304 ||style="background-color: #66ff66;" | 49 757 || style="background-color: #66ff66;" | 103 380 ||style="background-color: #66ff66;" | 196 689 ||style="background-color: #66ff66;" | 350 700 || style="background-color: #66ff66;" | 593 989
|-
| '''15''' ||style="background-color: #81BEF7;" | '''96''' ||style="background-color: yellow;" | '''448''' ||style="background-color: orange;" | 1 420 ||style="background-color: #66ff66;" | 4 292 ||style="background-color: #66ff66;" | 12 232 ||style="background-color: #66ff66;" | 32 092 ||style="background-color: #66ff66;" | 68 944 ||style="background-color: #66ff66;" | 142 516 ||style="background-color: #66ff66;" | 276 928 ||style="background-color: #66ff66;" | 514 580 ||style="background-color: #66ff66;" | 908 480
|-
| '''16''' ||style="background-color: #81BEF7;" | '''112''' ||style="background-color: orange;" | 518 ||style="background-color: #66ff66;" | 1 788 ||style="background-color: #66ff66;" | 5 847 ||style="background-color: #66ff66;" | 17 733 ||style="background-color: #66ff66;" | 44 328 ||style="background-color: #66ff66;" | 107 748 ||style="background-color: #66ff66;" | 232 245 ||style="background-color: #66ff66;" | 479 255 ||style="background-color: #66ff66;" | 924 420 ||style="background-color: #66ff66;" | 1 702 428
|-
| '''17''' ||style="background-color: orange;" | '''130''' ||style="background-color: gold;" | 570 ||style="background-color: gold;" | 1 954 ||style="background-color: #66ff66;" | 6 468 ||style="background-color: #66ff66;" | 20 360 ||style="background-color: #66ff66;" | 57 684 ||style="background-color: #66ff66;" | 136 512 ||style="background-color: #66ff66;" | 321 780 ||style="background-color: #66ff66;" | 659 464 ||style="background-color: #66ff66;" | 1 350 820 ||style="background-color: #66ff66;" | 2 479 104
|-
| '''18''' ||style="background-color: orange;" | '''138''' ||style="background-color: ;" | ||style="background-color: gold;" | 2 645 ||style="background-color: #66ff66;" | 8 248 ||style="background-color: #66ff66;" | 27 273 ||style="background-color: #66ff66;" | 77 577 ||style="background-color: #66ff66;" | 205 601 ||style="background-color: #66ff66;" | 483 523 ||style="background-color: #66ff66;" | 1 024 915 ||style="background-color: #66ff66;" | 2 202 955 ||style="background-color: #66ff66;" | 4 388 640
|-
| '''19''' ||style="background-color: orange;" | '''156''' ||style="background-color: ;" | ||style="background-color: ;" | ||style="background-color: #66ff66;" | 9 652 ||style="background-color: #66ff66;" | 29 928 ||style="background-color: #66ff66;" | 99 420 ||style="background-color: #66ff66;" | 257 144 ||style="background-color: #66ff66;" | 652 004 ||style="background-color: #66ff66;" | 1 388 608 ||style="background-color: #66ff66;" | 3 101 860 ||style="background-color: #66ff66;" | 6 037 496
|-
| '''20''' ||style="background-color: orange;" | '''171''' ||style="background-color: ;" | ||style="background-color: #66ff66;" | 2 828 ||style="background-color: #66ff66;" | 12 396 ||style="background-color: #66ff66;" | 39 733 ||style="background-color: #66ff66;" | 131 835 ||style="background-color: #66ff66;" | 358 089 ||style="background-color: #66ff66;" | 930 184 ||style="background-color: #66ff66;" | 2 232 648 ||style="background-color: #66ff66;" | 4 529 265 ||style="background-color: #66ff66;" | 10 001 820
|}
</center>

The following table is the key to the colors in the table presented above:

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: #ffffff; text-align: center;" | '''*''' || Numbers in '''bold''' indicate graphs known to be optimal.
|-
|style="background-color: #bbffff; text-align: center;" | * || Optimal graphs.
|-
|style="background-color: beige; text-align: center;" | * || Optimal graphs found by E. Monakhova.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graphs found by H. Macbeth, J. Šiagiová, J. Širáň and T. Vetrík.
|-
|style="background-color: magenta; text-align: center;" | * || Graphs found by R. Dougherty and V. Faber and independently for d=6 by E. Monakhova.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by B. McKay.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by R. Lewis.
|-
|style="background-color: #CCFF00; text-align: center;" | * || Graphs found by R. Lewis and independently by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, H. Pérez-Rosés and J. Ryan.
|-
|style="background-color: orange; text-align: center;" | * || Graphs found by D. Bevan, G. Erskine and R. Lewis.
|-
|style="background-color: gold; text-align: center;" | * || Graphs found by G. Erskine.
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by O. Monakhov and E. Monakhova.

|}
</center>

===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' || '''11''' || '''12'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|8
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|12
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|20
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|24
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|28
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|32
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|44
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|48
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|40
|-
|'''100%'''
|}
|-

|'''4'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|13
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|25
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|41
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|61
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|85
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|113
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|145
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|181
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|221
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|265
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|313
|-
|'''100%'''
|}
|-

|'''5'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|16
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|36
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|64
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|100
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|144
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|196
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|256
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|324
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|400
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|484
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|576
|-
|'''100%'''
|}
|-

|'''6'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|21
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|55
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|117
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|203
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|333
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|515
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|737
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 027
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 393
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 815
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 329
|-
|'''100%'''
|}
|-

|'''7'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|26
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|76
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|160
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|308
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|536
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|828
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 232
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 764
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|2 392
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 672
|-
|89%
|}
|-

|'''8'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|35
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|104
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|248
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|528
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|984
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 712
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 649
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 641
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 361
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|11 969
|-
|76%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|16 641
|-
|75%
|}
|-

|'''9'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|42
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|320
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|700
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|1 416
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 530
|-
|72%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 890
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|9 290
|-
|73%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14 002
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 330
|-
|74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 610
|-
|74%
|}
|-

|'''10'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|51
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|177
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|681
|-
|67%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 683
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|65%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7 183
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13 073
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22 363
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|36 365
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|56 695
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|85 305
|-
|62%
|}
|-

|'''11'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|56
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|210
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|912
|-
|63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2 364
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5 336
|-
|60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10 836
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|20 256
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|35 436
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|58 728
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93 060
|-
|61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|142 000
|-
|61%
|}
|-

|'''12'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|67
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|275
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 289
|-
|64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3 653
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8 989
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19 825
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|40 081
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|75 517
|-
|53%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|134 245
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|227 305
|-
|52%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|369 305
|-
|52%
|}
|-

|'''13'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|80
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|312
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1 666
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4 942
|-
|54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12 642
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|28 814
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|59 906
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|115 598
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|209 762
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|361 550
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|596610
|-
|50%
|}
|-

|'''14'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|90
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 381
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 241
|-
|55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 183
|-
|51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 19 825
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 48 639
|-
|48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 795 455
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 392 065
|-
|43%
|}
|-

|'''15'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 96
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 448
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 816
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 9 424
|-
|46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 27 008
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 68 464
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 157 184
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 332 688
|-
|43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 658 048
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 229 360
|-
|42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 187 520
|-
|42%
|}
|-

|'''16'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 112
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 833
|-
|62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 649
|-
|49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 13 073
|-
|45%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 40 081
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 108 545
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 265 729
|-
|41%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 485 825
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 673 345
|-
|36%
|}
|-

|'''17'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 130
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 978
|-
|58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|44%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 16 722
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 53 154
|-
|38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 148 626
|-
|39%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 374 274
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 864 146
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 854 882
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 742 290
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 159 170
|-
|35%
|}

|'''18'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 138
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 159
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 5 641
|-
|47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 22 363
|-
|37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 75 517
|-
|36%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 224 143
|-
|35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 598 417
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 462 563
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 7 059 735
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 14 218 905
|-
|31%
|}

|'''19'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 156
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 340
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 6 800
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 28 004
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 97 880
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 299 660
|-
|33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 822 560
|-
|31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 2 060 980
|-
|32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 780 008
|-
|29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 10 377 180
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 21 278 640
|-
|28%
|}

|'''20'''
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
| 171
|-
|'''100%'''
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 561
|-
|
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 361
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 36 365
|-
|34%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 134 245
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 433 905
|-
|30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 1 256 465
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 3 317 445
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 8 097 453
|-
|28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 18 474 633
|-
|25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 39 753 273
|-
|25%
|}

|}
</center>

==References==
* D. Bevan, G. Erskine, and R. Lewis. Large circulant graphs of fixed diameter and arbitrary degree. [http://arxiv.org/abs/1506.04962 ArXiv]
* R. Feria-Puron, J. Ryan, and H. Perez-Roses. Searching for Large Multi-Loop Networks. Electronic Notes in Discrete Mathematics, vol. 46 (2014), pp. 233-240. doi:10.1016/j.endm.2014.08.031. [http://www.sciencedirect.com/science/article/pii/S1571065314000328 Link to journal]
* R.R. Lewis. The Degree/Diameter Problem for Circulant Graphs of Degree 8 and 9. The Electronic Journal of Combinatorics, vol. 21(4) (2014), #P4.50. [http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p50 Link to journal]
* E.A. Monakhova, Synthesis of optimal Diophantine structures, Comput. Syst. Novosibirsk , 80 (1979), p.18--35. (in Russian).
* E. Monakhova, Optimal Triple Loop Networks with Given Transmission Delay: Topological Design and Routing, Inter. Network Optimization Conference, (INOC'2003), Evry/Paris, France, (2003), p.410--415.
* E.A. Monakhova . On synthesis of multidimensional circulant graphs of diameter two, Bulletin of the Tomsk Polytechnic University. 323(2) (2013), p.25--28. (in Russian). [http://izvestiya.tpu.ru/en/archive/article.html?id=265621&journalId= Link to journal]

The Degree Diameter Problem for Circulant Graphs

2019-06-07T13:51:06Z

Roblewis: /* Table of the orders of the largest known circulant graphs */

The Degree Diameter Problem for Circulant Graphs

2019-06-07T13:50:28Z

Roblewis: /* Table of the orders of the largest known circulant graphs */

The Degree Diameter Problem for Circulant Graphs

2019-06-07T13:48:52Z

Roblewis: /* Table of the orders of the largest known circulant graphs */

The Degree Diameter Problem for Circulant Graphs

2019-03-14T12:09:14Z

Roblewis:

The Degree Diameter Problem for Circulant Graphs

2019-03-08T20:08:23Z

Roblewis:

The Degree Diameter Problem for Circulant Graphs

2019-03-08T19:10:40Z

Roblewis:

The Degree Diameter Problem for Circulant Graphs

2019-03-08T19:03:25Z

Roblewis:

The Degree Diameter Problem for Circulant Graphs

2019-03-08T18:44:56Z

Roblewis: