Combinatorics Wiki - User contributions [en-gb]

Tables and Results

2025-05-13T09:05:04Z

Grahame: /* Table 2: bounds for cages of girth 5 */

==Trivalent cages==
===Table 1: known trivalent cages===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''g'' ''' || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''|| style="background-color: #cccccc;" | '''8'''|| style="background-color: #cccccc;" | '''9'''|| style="background-color: #cccccc;" | '''10'''|| style="background-color: #cccccc;" | '''11'''|| style="background-color: #cccccc;" | '''12'''
|-
|style="background-color: #cccccc;"| ''' ''n(3,g)'' ''' || 10 ||14 ||24 ||30 ||58 ||70 ||112 ||126
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 1|| 1|| 18|| 3|| 1|| 1
|}
</center>

===Table 2: bounds for trivalent cages===
Optimal graphs are marked in bold
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Girth ''g'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' ||style="background-color: #cccccc;"| ''' Number of cages''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"| 5 ||style="background-color: #cccccc;"| 10 || '''10''' ||1 ||[http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"| 6 ||style="background-color: #cccccc;"| 14 ||'''14'''|| 1 ||[http://en.wikipedia.org/wiki/Heawood_graph Heawood]
|-
|style="background-color: #cccccc;"| 7 ||style="background-color: #cccccc;"| 24 || '''24''' ||1 ||McGee
|-
|style="background-color: #cccccc;"| 8 ||style="background-color: #cccccc;"| 30 ||'''30'''|| 1 ||[http://en.wikipedia.org/wiki/Tutte_eight_cage Tutte]
|-
|style="background-color: #cccccc;"| 9 ||style="background-color: #cccccc;"| 58 ||'''58''' ||18 ||Brinkmann-McKay-Saager
|-
|style="background-color: #cccccc;"| 10 ||style="background-color: #cccccc;"| 70 ||'''70''' ||3 ||O’Keefe-Wong
|-
|style="background-color: #cccccc;"| 11 ||style="background-color: #cccccc;"| 112 || '''112'''|| 1 ||McKay-Myrvold; Balaban
|-
|style="background-color: #cccccc;"| 12 ||style="background-color: #cccccc;"| 126 || '''126'''|| 1|| Benson
|-
|style="background-color: #cccccc;"| 13 ||style="background-color: #cccccc;"| 202 ||272|| ||McKay-Myrvold; Hoare
|-
|style="background-color: #cccccc;"| 14 ||style="background-color: #cccccc;"| 258 ||384 ||||McKay; Exoo
|-
|style="background-color: #cccccc;"| 15 ||style="background-color: #cccccc;"| 384 ||620|||| Biggs
|-
|style="background-color: #cccccc;"| 16 ||style="background-color: #cccccc;"| 512 ||936|| ||Stubbe
|-
|style="background-color: #cccccc;"| 17 ||style="background-color: #cccccc;"| 768 ||2048|| ||Exoo
|-
|style="background-color: #cccccc;"| 18 ||style="background-color: #cccccc;"| 1024 ||2560|| ||Exoo
|-
|style="background-color: #cccccc;"| 19 ||style="background-color: #cccccc;"| 1536 ||4324|||| Hoare, H(47)
|-
|style="background-color: #cccccc;"| 20 ||style="background-color: #cccccc;"| 2048 || 5376 || ||Exoo
|-
|style="background-color: #cccccc;"| 21 ||style="background-color: #cccccc;"| 3072 ||16028 || ||Exoo
|-
|style="background-color: #cccccc;"| 22 ||style="background-color: #cccccc;"| 4096 || 16206|| || Biggs-Hoare, S(73)
|-
|style="background-color: #cccccc;"| 23 ||style="background-color: #cccccc;"| 6144 ||35446 || ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 24 ||style="background-color: #cccccc;"| 8192 || 35640 || ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 25 ||style="background-color: #cccccc;"| 12288 ||108906|| || Exoo
|-
|style="background-color: #cccccc;"| 26 ||style="background-color: #cccccc;"| 16384 || 109200 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 27 ||style="background-color: #cccccc;"| 24576 ||285852 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 28 ||style="background-color: #cccccc;"| 32768 ||368640|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 29 ||style="background-color: #cccccc;"| 49152 ||805746|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 30 ||style="background-color: #cccccc;"| 65536 ||806736|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 31 ||style="background-color: #cccccc;"| 98304 ||1440338|| ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 32 ||style="background-color: #cccccc;"| 131072 ||1441440|| || Erskine-Tuite
|}
</center>

==Cages of girth 5 and 6==
Optimal graphs are marked in bold.
===Table 1: known cages of Girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''k'' ''' || style="background-color: #cccccc;" | 3 || style="background-color: #cccccc;" | 4 || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''
|-
|style="background-color: #cccccc;"| ''' ''n(k,5)'' ''' || 10 || 19|| 30|| 40|| 50
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 4|| 1|| 1
|}
</center>

===Table 2: bounds for cages of girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"|10 ||'''10'''|| [http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|19 ||'''19'''|| Robertson
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|30 ||'''30'''|| Robertson-Wegner-Wong
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|40 ||'''40'''|| Wong
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|50||''' 50'''|| [http://en.wikipedia.org/wiki/Hoffman-Singleton_graph Hoﬀman-Singleton]
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|67|| 80|| Royle
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|86|| 96 ||Jørgensen
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|103|| 124 ||Exoo
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|124 ||154|| Exoo
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|147|| 203|| Exoo
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|174 ||230 ||Exoo
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|199 ||288 ||Jørgensen
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|230 ||312 ||Jørgensen
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|259 ||336|| Jørgensen
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|294|| 448 ||Schwenk
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|327|| 480|| Schwenk
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|364|| 512|| Schwenk
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|403|| 576 ||Jørgensen
|}
</center>

===Table 3: bounds for cages of girth 6===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"| 14 ||'''14'''|| [http://en.wikipedia.org/wiki/Projective_plane Projective Plane]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|26 ||'''26'''|| Projective Plane
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|42 ||'''42'''|| Projective Plane
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|62 ||'''62'''|| Projective Plane
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|90 ||'''90'''|| O’Keefe-Wong
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|114 ||'''114'''|| Projective Plane
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|146 ||'''146'''|| Projective Plane
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|182 ||'''182'''|| Projective Plane
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|224 ||240|| Wong
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|266||'''266'''|| Projective Plane
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|314 ||336|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|366 ||'''366'''|| Projective Plane
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|422 ||462|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|482 ||504|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|546 ||'''546'''|| Projective Plane
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|614 ||'''614'''|| Projective Plane
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|686 ||720 ||Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|762 ||'''762'''|| Projective Plane
|}
</center>

[[Category:The Cage Problem]]

Tables and Results

2025-05-12T22:30:11Z

Grahame: /* Table 2: bounds for cages of girth 5 */

==Trivalent cages==
===Table 1: known trivalent cages===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''g'' ''' || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''|| style="background-color: #cccccc;" | '''8'''|| style="background-color: #cccccc;" | '''9'''|| style="background-color: #cccccc;" | '''10'''|| style="background-color: #cccccc;" | '''11'''|| style="background-color: #cccccc;" | '''12'''
|-
|style="background-color: #cccccc;"| ''' ''n(3,g)'' ''' || 10 ||14 ||24 ||30 ||58 ||70 ||112 ||126
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 1|| 1|| 18|| 3|| 1|| 1
|}
</center>

===Table 2: bounds for trivalent cages===
Optimal graphs are marked in bold
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Girth ''g'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' ||style="background-color: #cccccc;"| ''' Number of cages''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"| 5 ||style="background-color: #cccccc;"| 10 || '''10''' ||1 ||[http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"| 6 ||style="background-color: #cccccc;"| 14 ||'''14'''|| 1 ||[http://en.wikipedia.org/wiki/Heawood_graph Heawood]
|-
|style="background-color: #cccccc;"| 7 ||style="background-color: #cccccc;"| 24 || '''24''' ||1 ||McGee
|-
|style="background-color: #cccccc;"| 8 ||style="background-color: #cccccc;"| 30 ||'''30'''|| 1 ||[http://en.wikipedia.org/wiki/Tutte_eight_cage Tutte]
|-
|style="background-color: #cccccc;"| 9 ||style="background-color: #cccccc;"| 58 ||'''58''' ||18 ||Brinkmann-McKay-Saager
|-
|style="background-color: #cccccc;"| 10 ||style="background-color: #cccccc;"| 70 ||'''70''' ||3 ||O’Keefe-Wong
|-
|style="background-color: #cccccc;"| 11 ||style="background-color: #cccccc;"| 112 || '''112'''|| 1 ||McKay-Myrvold; Balaban
|-
|style="background-color: #cccccc;"| 12 ||style="background-color: #cccccc;"| 126 || '''126'''|| 1|| Benson
|-
|style="background-color: #cccccc;"| 13 ||style="background-color: #cccccc;"| 202 ||272|| ||McKay-Myrvold; Hoare
|-
|style="background-color: #cccccc;"| 14 ||style="background-color: #cccccc;"| 258 ||384 ||||McKay; Exoo
|-
|style="background-color: #cccccc;"| 15 ||style="background-color: #cccccc;"| 384 ||620|||| Biggs
|-
|style="background-color: #cccccc;"| 16 ||style="background-color: #cccccc;"| 512 ||936|| ||Stubbe
|-
|style="background-color: #cccccc;"| 17 ||style="background-color: #cccccc;"| 768 ||2048|| ||Exoo
|-
|style="background-color: #cccccc;"| 18 ||style="background-color: #cccccc;"| 1024 ||2560|| ||Exoo
|-
|style="background-color: #cccccc;"| 19 ||style="background-color: #cccccc;"| 1536 ||4324|||| Hoare, H(47)
|-
|style="background-color: #cccccc;"| 20 ||style="background-color: #cccccc;"| 2048 || 5376 || ||Exoo
|-
|style="background-color: #cccccc;"| 21 ||style="background-color: #cccccc;"| 3072 ||16028 || ||Exoo
|-
|style="background-color: #cccccc;"| 22 ||style="background-color: #cccccc;"| 4096 || 16206|| || Biggs-Hoare, S(73)
|-
|style="background-color: #cccccc;"| 23 ||style="background-color: #cccccc;"| 6144 ||35446 || ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 24 ||style="background-color: #cccccc;"| 8192 || 35640 || ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 25 ||style="background-color: #cccccc;"| 12288 ||108906|| || Exoo
|-
|style="background-color: #cccccc;"| 26 ||style="background-color: #cccccc;"| 16384 || 109200 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 27 ||style="background-color: #cccccc;"| 24576 ||285852 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 28 ||style="background-color: #cccccc;"| 32768 ||368640|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 29 ||style="background-color: #cccccc;"| 49152 ||805746|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 30 ||style="background-color: #cccccc;"| 65536 ||806736|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 31 ||style="background-color: #cccccc;"| 98304 ||1440338|| ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 32 ||style="background-color: #cccccc;"| 131072 ||1441440|| || Erskine-Tuite
|}
</center>

==Cages of girth 5 and 6==
Optimal graphs are marked in bold.
===Table 1: known cages of Girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''k'' ''' || style="background-color: #cccccc;" | 3 || style="background-color: #cccccc;" | 4 || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''
|-
|style="background-color: #cccccc;"| ''' ''n(k,5)'' ''' || 10 || 19|| 30|| 40|| 50
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 4|| 1|| 1
|}
</center>

===Table 2: bounds for cages of girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"|10 ||'''10'''|| [http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|19 ||'''19'''|| Robertson
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|30 ||'''30'''|| Robertson-Wegner-Wong
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|40 ||'''40'''|| Wong
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|50||''' 50'''|| [http://en.wikipedia.org/wiki/Hoffman-Singleton_graph Hoﬀman-Singleton]
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|67|| 80|| Royle
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|86|| 96 ||Jørgensen
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|103|| 124 ||Exoo
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|124 ||154|| Exoo
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|147|| 203|| Exoo
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|174 ||240 ||Exoo
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|199 ||288 ||Jørgensen
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|230 ||312 ||Jørgensen
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|259 ||336|| Jørgensen
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|294|| 448 ||Schwenk
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|327|| 480|| Schwenk
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|364|| 512|| Schwenk
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|403|| 576 ||Jørgensen
|}
</center>

===Table 3: bounds for cages of girth 6===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"| 14 ||'''14'''|| [http://en.wikipedia.org/wiki/Projective_plane Projective Plane]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|26 ||'''26'''|| Projective Plane
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|42 ||'''42'''|| Projective Plane
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|62 ||'''62'''|| Projective Plane
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|90 ||'''90'''|| O’Keefe-Wong
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|114 ||'''114'''|| Projective Plane
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|146 ||'''146'''|| Projective Plane
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|182 ||'''182'''|| Projective Plane
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|224 ||240|| Wong
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|266||'''266'''|| Projective Plane
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|314 ||336|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|366 ||'''366'''|| Projective Plane
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|422 ||462|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|482 ||504|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|546 ||'''546'''|| Projective Plane
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|614 ||'''614'''|| Projective Plane
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|686 ||720 ||Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|762 ||'''762'''|| Projective Plane
|}
</center>

[[Category:The Cage Problem]]

Tables and Results

2025-05-02T17:10:48Z

Grahame: /* Table 2: bounds for trivalent cages */

==Trivalent cages==
===Table 1: known trivalent cages===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''g'' ''' || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''|| style="background-color: #cccccc;" | '''8'''|| style="background-color: #cccccc;" | '''9'''|| style="background-color: #cccccc;" | '''10'''|| style="background-color: #cccccc;" | '''11'''|| style="background-color: #cccccc;" | '''12'''
|-
|style="background-color: #cccccc;"| ''' ''n(3,g)'' ''' || 10 ||14 ||24 ||30 ||58 ||70 ||112 ||126
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 1|| 1|| 18|| 3|| 1|| 1
|}
</center>

===Table 2: bounds for trivalent cages===
Optimal graphs are marked in bold
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Girth ''g'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' ||style="background-color: #cccccc;"| ''' Number of cages''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"| 5 ||style="background-color: #cccccc;"| 10 || '''10''' ||1 ||[http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"| 6 ||style="background-color: #cccccc;"| 14 ||'''14'''|| 1 ||[http://en.wikipedia.org/wiki/Heawood_graph Heawood]
|-
|style="background-color: #cccccc;"| 7 ||style="background-color: #cccccc;"| 24 || '''24''' ||1 ||McGee
|-
|style="background-color: #cccccc;"| 8 ||style="background-color: #cccccc;"| 30 ||'''30'''|| 1 ||[http://en.wikipedia.org/wiki/Tutte_eight_cage Tutte]
|-
|style="background-color: #cccccc;"| 9 ||style="background-color: #cccccc;"| 58 ||'''58''' ||18 ||Brinkmann-McKay-Saager
|-
|style="background-color: #cccccc;"| 10 ||style="background-color: #cccccc;"| 70 ||'''70''' ||3 ||O’Keefe-Wong
|-
|style="background-color: #cccccc;"| 11 ||style="background-color: #cccccc;"| 112 || '''112'''|| 1 ||McKay-Myrvold; Balaban
|-
|style="background-color: #cccccc;"| 12 ||style="background-color: #cccccc;"| 126 || '''126'''|| 1|| Benson
|-
|style="background-color: #cccccc;"| 13 ||style="background-color: #cccccc;"| 202 ||272|| ||McKay-Myrvold; Hoare
|-
|style="background-color: #cccccc;"| 14 ||style="background-color: #cccccc;"| 258 ||384 ||||McKay; Exoo
|-
|style="background-color: #cccccc;"| 15 ||style="background-color: #cccccc;"| 384 ||620|||| Biggs
|-
|style="background-color: #cccccc;"| 16 ||style="background-color: #cccccc;"| 512 ||936|| ||Stubbe
|-
|style="background-color: #cccccc;"| 17 ||style="background-color: #cccccc;"| 768 ||2048|| ||Exoo
|-
|style="background-color: #cccccc;"| 18 ||style="background-color: #cccccc;"| 1024 ||2560|| ||Exoo
|-
|style="background-color: #cccccc;"| 19 ||style="background-color: #cccccc;"| 1536 ||4324|||| Hoare, H(47)
|-
|style="background-color: #cccccc;"| 20 ||style="background-color: #cccccc;"| 2048 || 5376 || ||Exoo
|-
|style="background-color: #cccccc;"| 21 ||style="background-color: #cccccc;"| 3072 ||16028 || ||Exoo
|-
|style="background-color: #cccccc;"| 22 ||style="background-color: #cccccc;"| 4096 || 16206|| || Biggs-Hoare, S(73)
|-
|style="background-color: #cccccc;"| 23 ||style="background-color: #cccccc;"| 6144 ||35446 || ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 24 ||style="background-color: #cccccc;"| 8192 || 35640 || ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 25 ||style="background-color: #cccccc;"| 12288 ||108906|| || Exoo
|-
|style="background-color: #cccccc;"| 26 ||style="background-color: #cccccc;"| 16384 || 109200 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 27 ||style="background-color: #cccccc;"| 24576 ||285852 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 28 ||style="background-color: #cccccc;"| 32768 ||368640|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 29 ||style="background-color: #cccccc;"| 49152 ||805746|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 30 ||style="background-color: #cccccc;"| 65536 ||806736|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 31 ||style="background-color: #cccccc;"| 98304 ||1440338|| ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 32 ||style="background-color: #cccccc;"| 131072 ||1441440|| || Erskine-Tuite
|}
</center>

==Cages of girth 5 and 6==
Optimal graphs are marked in bold.
===Table 1: known cages of Girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''k'' ''' || style="background-color: #cccccc;" | 3 || style="background-color: #cccccc;" | 4 || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''
|-
|style="background-color: #cccccc;"| ''' ''n(k,5)'' ''' || 10 || 19|| 30|| 40|| 50
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 4|| 1|| 1
|}
</center>

===Table 2: bounds for cages of girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"|10 ||'''10'''|| [http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|19 ||'''19'''|| Robertson
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|30 ||'''30'''|| Robertson-Wegner-Wong
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|40 ||'''40'''|| Wong
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|50||''' 50'''|| [http://en.wikipedia.org/wiki/Hoffman-Singleton_graph Hoﬀman-Singleton]
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|67|| 80|| Royle
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|86|| 96 ||Jørgensen
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|103|| 126 ||Exoo
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|124 ||156|| Jørgensen
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|147|| 203|| Exoo
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|174 ||240 ||Exoo
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|199 ||288 ||Jørgensen
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|230 ||312 ||Jørgensen
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|259 ||336|| Jørgensen
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|294|| 448 ||Schwenk
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|327|| 480|| Schwenk
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|364|| 512|| Schwenk
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|403|| 576 ||Jørgensen
|}
</center>

===Table 3: bounds for cages of girth 6===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"| 14 ||'''14'''|| [http://en.wikipedia.org/wiki/Projective_plane Projective Plane]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|26 ||'''26'''|| Projective Plane
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|42 ||'''42'''|| Projective Plane
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|62 ||'''62'''|| Projective Plane
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|90 ||'''90'''|| O’Keefe-Wong
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|114 ||'''114'''|| Projective Plane
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|146 ||'''146'''|| Projective Plane
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|182 ||'''182'''|| Projective Plane
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|224 ||240|| Wong
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|266||'''266'''|| Projective Plane
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|314 ||336|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|366 ||'''366'''|| Projective Plane
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|422 ||462|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|482 ||504|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|546 ||'''546'''|| Projective Plane
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|614 ||'''614'''|| Projective Plane
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|686 ||720 ||Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|762 ||'''762'''|| Projective Plane
|}
</center>

[[Category:The Cage Problem]]

Tables and Results

2025-03-20T17:59:13Z

Grahame: /* Table 3: bounds for cages of girth 6 */

==Trivalent cages==
===Table 1: known trivalent cages===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''g'' ''' || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''|| style="background-color: #cccccc;" | '''8'''|| style="background-color: #cccccc;" | '''9'''|| style="background-color: #cccccc;" | '''10'''|| style="background-color: #cccccc;" | '''11'''|| style="background-color: #cccccc;" | '''12'''
|-
|style="background-color: #cccccc;"| ''' ''n(3,g)'' ''' || 10 ||14 ||24 ||30 ||58 ||70 ||112 ||126
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 1|| 1|| 18|| 3|| 1|| 1
|}
</center>

===Table 2: bounds for trivalent cages===
Optimal graphs are marked in bold
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Girth ''g'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' ||style="background-color: #cccccc;"| ''' Number of cages''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"| 5 ||style="background-color: #cccccc;"| 10 || '''10''' ||1 ||[http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"| 6 ||style="background-color: #cccccc;"| 14 ||'''14'''|| 1 ||[http://en.wikipedia.org/wiki/Heawood_graph Heawood]
|-
|style="background-color: #cccccc;"| 7 ||style="background-color: #cccccc;"| 24 || '''24''' ||1 ||McGee
|-
|style="background-color: #cccccc;"| 8 ||style="background-color: #cccccc;"| 30 ||'''30'''|| 1 ||[http://en.wikipedia.org/wiki/Tutte_eight_cage Tutte]
|-
|style="background-color: #cccccc;"| 9 ||style="background-color: #cccccc;"| 58 ||'''58''' ||18 ||Brinkmann-McKay-Saager
|-
|style="background-color: #cccccc;"| 10 ||style="background-color: #cccccc;"| 70 ||'''70''' ||3 ||O’Keefe-Wong
|-
|style="background-color: #cccccc;"| 11 ||style="background-color: #cccccc;"| 112 || '''112'''|| 1 ||McKay-Myrvold; Balaban
|-
|style="background-color: #cccccc;"| 12 ||style="background-color: #cccccc;"| 126 || '''126'''|| 1|| Benson
|-
|style="background-color: #cccccc;"| 13 ||style="background-color: #cccccc;"| 202 ||272|| ||McKay-Myrvold; Hoare
|-
|style="background-color: #cccccc;"| 14 ||style="background-color: #cccccc;"| 258 ||384 ||||McKay; Exoo
|-
|style="background-color: #cccccc;"| 15 ||style="background-color: #cccccc;"| 384 ||620|||| Biggs
|-
|style="background-color: #cccccc;"| 16 ||style="background-color: #cccccc;"| 512 ||960|| ||Exoo
|-
|style="background-color: #cccccc;"| 17 ||style="background-color: #cccccc;"| 768 ||2176|| ||Exoo
|-
|style="background-color: #cccccc;"| 18 ||style="background-color: #cccccc;"| 1024 ||2560|| ||Exoo
|-
|style="background-color: #cccccc;"| 19 ||style="background-color: #cccccc;"| 1536 ||4324|||| Hoare, H(47)
|-
|style="background-color: #cccccc;"| 20 ||style="background-color: #cccccc;"| 2048 || 5376 || ||Exoo
|-
|style="background-color: #cccccc;"| 21 ||style="background-color: #cccccc;"| 3072 ||16028 || ||Exoo
|-
|style="background-color: #cccccc;"| 22 ||style="background-color: #cccccc;"| 4096 || 16206|| || Biggs-Hoare, S(73)
|-
|style="background-color: #cccccc;"| 23 ||style="background-color: #cccccc;"| 6144 ||35446 || ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 24 ||style="background-color: #cccccc;"| 8192 || 35640 || ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 25 ||style="background-color: #cccccc;"| 12288 ||108906|| || Exoo
|-
|style="background-color: #cccccc;"| 26 ||style="background-color: #cccccc;"| 16384 || 109200 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 27 ||style="background-color: #cccccc;"| 24576 ||285852 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 28 ||style="background-color: #cccccc;"| 32768 ||368640|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 29 ||style="background-color: #cccccc;"| 49152 ||805746|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 30 ||style="background-color: #cccccc;"| 65536 ||806736|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 31 ||style="background-color: #cccccc;"| 98304 ||1440338|| ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 32 ||style="background-color: #cccccc;"| 131072 ||1441440|| || Erskine-Tuite
|}
</center>

==Cages of girth 5 and 6==
Optimal graphs are marked in bold.
===Table 1: known cages of Girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''k'' ''' || style="background-color: #cccccc;" | 3 || style="background-color: #cccccc;" | 4 || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''
|-
|style="background-color: #cccccc;"| ''' ''n(k,5)'' ''' || 10 || 19|| 30|| 40|| 50
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 4|| 1|| 1
|}
</center>

===Table 2: bounds for cages of girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"|10 ||'''10'''|| [http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|19 ||'''19'''|| Robertson
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|30 ||'''30'''|| Robertson-Wegner-Wong
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|40 ||'''40'''|| Wong
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|50||''' 50'''|| [http://en.wikipedia.org/wiki/Hoffman-Singleton_graph Hoﬀman-Singleton]
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|67|| 80|| Royle
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|86|| 96 ||Jørgensen
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|103|| 126 ||Exoo
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|124 ||156|| Jørgensen
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|147|| 203|| Exoo
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|174 ||240 ||Exoo
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|199 ||288 ||Jørgensen
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|230 ||312 ||Jørgensen
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|259 ||336|| Jørgensen
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|294|| 448 ||Schwenk
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|327|| 480|| Schwenk
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|364|| 512|| Schwenk
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|403|| 576 ||Jørgensen
|}
</center>

===Table 3: bounds for cages of girth 6===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"| 14 ||'''14'''|| [http://en.wikipedia.org/wiki/Projective_plane Projective Plane]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|26 ||'''26'''|| Projective Plane
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|42 ||'''42'''|| Projective Plane
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|62 ||'''62'''|| Projective Plane
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|90 ||'''90'''|| O’Keefe-Wong
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|114 ||'''114'''|| Projective Plane
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|146 ||'''146'''|| Projective Plane
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|182 ||'''182'''|| Projective Plane
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|224 ||240|| Wong
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|266||'''266'''|| Projective Plane
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|314 ||336|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|366 ||'''366'''|| Projective Plane
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|422 ||462|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|482 ||504|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|546 ||'''546'''|| Projective Plane
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|614 ||'''614'''|| Projective Plane
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|686 ||720 ||Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|762 ||'''762'''|| Projective Plane
|}
</center>

[[Category:The Cage Problem]]

TempArcTransitive

2024-10-02T17:49:16Z

Grahame:

'''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;" | '''10''' || style="background-color: #eee;" | '''14''' || style="background-color: #eee;" | '''30''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | '''64''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''234''' ||style="background-color: #eee;" | '''364''' ||style="background-color: #eee;" | '''1250'''
|-
| '''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

2024-09-30T12:25:49Z

Grahame: Created page with "'''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===..."

'''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;" | 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>

Open problems in degree/girth

2023-07-16T08:34:49Z

Grahame:

This page lists a number of open problems in the degree/girth (cage) problem.

For the background and history of the problem, see G. Exoo and R. Jajcay, ''Dynamic cage survey'', Electron. J. Combin. DS16 (2012).

===Girth 5 graphs===
The current best known constructions for degree <math>d</math> and girth 5 in the undirected cage problem have order <math>2d^2+o(d^2)</math>. (See the survey for details.) This is asymptotically the same as the best graphs of girth 6. Is it possible to construct an infinite family of graphs of degree <math>d</math> and girth 5 with order <math>cd^2+o(d^2)</math> where <math>c<2</math> is a constant?

Expected difficulty: hard.

Open problems in degree/girth

2023-07-16T08:17:41Z

Grahame:

This page lists a number of open problems in the degree/girth (cage) problem.

For the background and history of the problem, see G. Exoo and R. Jajcay, ''Dynamic cage survey'', Electron. J. Combin. DS16 (2012).

===Girth 5 graphs===
The current best known constructions for degree <math>d</math> and girth 5 in the undirected cage problem have order <math>2d^2+o(d)</math>. (See the survey for details.) This is asymptotically the same as the best graphs of girth 6. Is it possible to construct an infinite family of graphs of degree <math>d</math> and girth 5 with order <math>cd^2+o(d)</math> where <math>c<2</math> is a constant?

Expected difficulty: hard.

Open problems in degree/diameter

2023-07-16T08:12:11Z

Grahame:

This page lists a number of open problems in the degree/diameter problem.

For the background and history of the problem, see M. Miller and J. Širáň, ''Moore graphs and beyond: a survey of the degree/diameter problem'', Electron. J. Combin. DS14 (2013).

Open problems in degree/girth

2023-07-16T08:07:06Z

Grahame: Created page with "This page lists a number of open problems in the degree/girth (cage) problem. For the background and history of the problem, see G. Exoo and R. Jajcay, Dynamic cage survey, E..."

This page lists a number of open problems in the degree/girth (cage) problem.

For the background and history of the problem, see G. Exoo and R. Jajcay, Dynamic cage survey, Electron. J. Combin. DS16 (2012).

===Girth 5 graphs===
The current best known constructions for degree <math>d</math> and girth 5

Open problems in degree/diameter

2023-07-16T07:58:44Z

Grahame: Created page with "This page lists a number of open problems in the degree/diameter problem."

This page lists a number of open problems in the degree/diameter problem.

Main Page

2023-07-16T07:57:39Z

Grahame:

==About Combinatorics Wiki==

Combinatorics Wiki is a wiki presenting the latest results on problems in various topics in the field of [http://en.wikipedia.org/wiki/Combinatorics combinatorics]. Combinatorics Wiki will only allow updates by active expert researchers in their fields, with the following goals:

* Creating a stable venue for researchers to announce published and pre-published work in real time. As many of the existing problems, in particular in extremal theory are of highly competitive nature, where new results very often supersede existing results, an up to date resource listing the most current results is therefore essential to the community working in a specific field. Taking into account the long time it can take to publish mathematical papers, it can be very helpful to announce and briefly describe new findings before the actual publication.

* Creating an extensive peer-reviewed source of information, allowing for new and existing researchers to stay up to date with work done by others in their field.

* Keeping a detailed history of previous work, findings, publications and results, in a simple user friendly wiki format.

* Allowing registered users to review and comment on unpublished and published work by other users.

* Giving supervisors and students ideas for new projects and open problems.

* Creating a stable community of researchers in different areas, and promoting collaborations.

==Citation==

If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.

* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). Combinatorics Wiki, http://combinatoricswiki.org.

If you are using a specific page in combinatoricsWiki, say the "The degree-diameter problem" page, then it would be better to cite the page as follows.

* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). The degree-diameter problem, Combinatorics Wiki, http://combinatoricswiki.org.

==List of problem areas==

* [[Enumeration of latin squares and rectangles]]

* [[The_Cage_Problem|The cage Problem or The degree/girth problem]]

* [[The Degree/Diameter Problem]]

* [[The maximum degree-and-diameter-bounded subgraph problem]]

* [[Extremal C_t-free graphs]]

==List of open problems==

Some open problems in areas covered by CombinatoricsWiki. When adding new problems, it would be helpful to indicate the likely level of difficulty and whether the problem might be suitable for research students.

* [[Open problems in degree/diameter]]

* [[Open problems in degree/girth]]

==List of video channels==

* [[Lectures Available Online|Lectures available online]]

* [[Documentaries Available Online|Documentaries available online]]

==Supporting organizations==

* [http://combinatorics-australasia.org/ CMSA - Combinatorial Mathematics Society of Australasia]

* [http://graphtheorygroup.com GTA - Graph Theory and Applications - The University of Newcastle, Australia]

* [http://www.indstate.edu/home.htm Indiana State University]

* [http://www-mat.upc.es/grup_de_grafs Research Group on Graph Theory and Combinatorics - UPC, Spain]

==Newsletters==

[[Combinatorial_Mathematics_Society_of_Australasia_Newsletters|Newsletters of the Combinatorial Mathematics Society of Australasia]]

==Meetings, seminars and talks==

Links to upcoming talks which may be of interest to researchers in the problem areas covered by Combinatorics Wiki.

[[Mirka_Miller%27s_Combinatorics_Webinar_Series|Mirka Miller's Combinatorics Webinar Series]]

==Employment Opportunities==

Members of the Combinatorics Wiki community are welcome to [[Employment Opportunities| advertise research internships, postdoc positions and other research and teaching openings]] in their respective institutions and others.

==Combinatorics Wiki rules==

Please read our rules of [[Rules and Regulations|use of Combinatorics Wiki]].

==Note for potential new contributors and moderators==

We are always interested in extending our list of problem areas. Please contact our [[List of Moderators|moderators]] with new ideas and suggestions. New registered users, '''[[Help:Editing|editing help can be found here]]''' (including adding pages, using mathematical formulas and embedding videos).

Temp VertexTransitive

2023-07-16T07:49:52Z

Grahame:

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

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

Below is the table of the largest known vertex-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: #f0f;" | '''10''' || style="background-color: #eee;" | '''14''' || style="background-color: #ff0;" | '''30''' ||style="background-color: #eee;" | '''60''' || style="background-color: #ff0;" | '''82''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #ff0;" | 546 ||style="background-color: #ff0;" | 1 250
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #8f0;" | 35 ||style="background-color: #eee;" | 84 || style="background-color: #8f0;" | 273 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | 18 ||style="background-color: #eee;" | 60 ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" | '''32''' ||style="background-color: #eee;" | 108 ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #0ff;" |'''50''' ||style="background-color: #eee;" |168 ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" | 48 ||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | 60 || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | 72 || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | 84 ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | 96 ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #f80;" | 162 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #f80;" | 338 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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;" | * || Cayley graphs; see the [[The_Degree_Diameter_Problem_for_Cayley_Graphs | separate page]] for details.
|-
|style="background-color: #f0f; text-align: center;" | * || The [https://mathworld.wolfram.com/PetersenGraph.html Petersen graph].
|-
|style="background-color: #ff0; text-align: center;" | * || See P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|-
|style="background-color: #0ff; text-align: center;" | * || The [https://mathworld.wolfram.com/Hoffman-SingletonGraph.html Hoffman-Singleton graph].
|-
|style="background-color: #f80; text-align: center;" | * || See B. D. McKay, M. Miller and J. Širáň, ''A note on large graphs of diameter two and given maximum degree''.
|-
|style="background-color: #8f0; text-align: center;" | * || See P. Potočnik, P. Spiga and G. Verret, ''Bounding the order of the vertex-stabiliser in 3-valent vertex transitive and 4-valent arc-transitive graphs''.
|}
</center>

Temp VertexTransitive

2022-02-24T18:00:20Z

Grahame: /* Table of the orders of the largest known vertex-transitive graphs for the undirected degree diameter problem */

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

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

Below is the table of the largest known vertex-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: #f0f;" | '''10''' || style="background-color: #eee;" | '''14''' || style="background-color: #ff0;" | '''30''' ||style="background-color: #eee;" | '''60''' || style="background-color: #ff0;" | '''82''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #ff0;" | 546 ||style="background-color: #ff0;" | 1 250
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #8f0;" | 35 ||style="background-color: #eee;" | 84 || style="background-color: #8f0;" | 273 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | 18 ||style="background-color: #eee;" | 60 ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" | '''32''' ||style="background-color: #eee;" | 108 ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #0ff;" |'''50''' ||style="background-color: #eee;" |168 ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" | 48 ||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | 60 || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | 72 || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | 84 ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | 96 ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #f80;" | 162 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #f80;" | 338 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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;" | * || Cayley graphs; see the [[The_Degree_Diameter_Problem_for_Cayley_Graphs | separate page]] for details.
|-
|style="background-color: #f0f; text-align: center;" | * || The [https://mathworld.wolfram.com/PetersenGraph.html Petersen graph].
|-
|style="background-color: #ff0; text-align: center;" | * || See P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|-
|style="background-color: #0ff; text-align: center;" | * || The [https://mathworld.wolfram.com/Hoffman-SingletonGraph.html Hoffman-Singleton graph].
|-
|style="background-color: #f80; text-align: center;" | * || See B. D. McKay, M. Miller and J. Širáň, ''A note on large graphs of diameter two and given maximum degree''.
|-
|style="background-color: #8f0; text-align: center;" | * || See P. Potočnik, P. Spiga and G. Verret, ''Bounding the order of the vertex-stabiliser in 3-valent vertex transitive and 4-valent arc-transitive graphs''.
|}
</center>

Temp VertexTransitive

2022-02-24T13:54:43Z

Grahame: /* Table of the orders of the largest known vertex-transitive graphs for the undirected degree diameter problem */

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

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

Below is the table of the largest known vertex-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: #f0f;" | '''10''' || style="background-color: #eee;" | '''14''' || style="background-color: #ff0;" | '''30''' ||style="background-color: #eee;" | '''60''' || style="background-color: #ff0;" | '''82''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #ff0;" | 546 ||style="background-color: #ff0;" | 1 250
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #eee;" | 30 ||style="background-color: #eee;" | 84 || style="background-color: #eee;" | 216 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | 18 ||style="background-color: #eee;" | 60 ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" | '''32''' ||style="background-color: #eee;" | 108 ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #0ff;" |'''50''' ||style="background-color: #eee;" |168 ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" | 48 ||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | 60 || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | 72 || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | 84 ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | 96 ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #f80;" | 162 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #f80;" | 338 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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;" | * || Cayley graphs; see the [[The_Degree_Diameter_Problem_for_Cayley_Graphs | separate page]] for details.
|-
|style="background-color: #f0f; text-align: center;" | * || The [https://mathworld.wolfram.com/PetersenGraph.html Petersen graph].
|-
|style="background-color: #ff0; text-align: center;" | * || See P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|-
|style="background-color: #0ff; text-align: center;" | * || The [https://mathworld.wolfram.com/Hoffman-SingletonGraph.html Hoffman-Singleton graph].
|-
|style="background-color: #f80; text-align: center;" | * || See B. D. McKay, M. Miller and J. Širáň, ''A note on large graphs of diameter two and given maximum degree''.
|}
</center>

Main Page

2022-02-24T13:27:53Z

Grahame: /* List of problem areas */

==About Combinatorics Wiki==

Combinatorics Wiki is a wiki presenting the latest results on problems in various topics in the field of [http://en.wikipedia.org/wiki/Combinatorics combinatorics]. Combinatorics Wiki will only allow updates by active expert researchers in their fields, with the following goals:

* Creating a stable venue for researchers to announce published and pre-published work in real time. As many of the existing problems, in particular in extremal theory are of highly competitive nature, where new results very often supersede existing results, an up to date resource listing the most current results is therefore essential to the community working in a specific field. Taking into account the long time it can take to publish mathematical papers, it can be very helpful to announce and briefly describe new findings before the actual publication.

* Creating an extensive peer-reviewed source of information, allowing for new and existing researchers to stay up to date with work done by others in their field.

* Keeping a detailed history of previous work, findings, publications and results, in a simple user friendly wiki format.

* Allowing registered users to review and comment on unpublished and published work by other users.

* Giving supervisors and students ideas for new projects and open problems.

* Creating a stable community of researchers in different areas, and promoting collaborations.

==Citation==

If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.

* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). Combinatorics Wiki, http://combinatoricswiki.org.

If you are using a specific page in combinatoricsWiki, say the "The degree-diameter problem" page, then it would be better to cite the page as follows.

* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). The degree-diameter problem, Combinatorics Wiki, http://combinatoricswiki.org.

==List of problem areas==

* [[Enumeration of latin squares and rectangles]]

* [[The_Cage_Problem|The cage Problem or The degree/girth problem]]

* [[The Degree/Diameter Problem]]

* [[The maximum degree-and-diameter-bounded subgraph problem]]

* [[Extremal C_t-free graphs]]

==List of video channels==

* [[Lectures Available Online|Lectures available online]]

* [[Documentaries Available Online|Documentaries available online]]

==Supporting organizations==

* [http://combinatorics-australasia.org/ CMSA - Combinatorial Mathematics Society of Australasia]

* [http://graphtheorygroup.com GTA - Graph Theory and Applications - The University of Newcastle, Australia]

* [http://www.indstate.edu/home.htm Indiana State University]

* [http://www-mat.upc.es/grup_de_grafs Research Group on Graph Theory and Combinatorics - UPC, Spain]

==Newsletters==

[[Combinatorial_Mathematics_Society_of_Australasia_Newsletters|Newsletters of the Combinatorial Mathematics Society of Australasia]]

==Meetings, seminars and talks==

Links to upcoming talks which may be of interest to researchers in the problem areas covered by Combinatorics Wiki.

[[Mirka_Miller%27s_Combinatorics_Webinar_Series|Mirka Miller's Combinatorics Webinar Series]]

==Employment Opportunities==

Members of the Combinatorics Wiki community are welcome to [[Employment Opportunities| advertise research internships, postdoc positions and other research and teaching openings]] in their respective institutions and others.

==Combinatorics Wiki rules==

Please read our rules of [[Rules and Regulations|use of Combinatorics Wiki]].

==Note for potential new contributors and moderators==

We are always interested in extending our list of problem areas. Please contact our [[List of Moderators|moderators]] with new ideas and suggestions. New registered users, '''[[Help:Editing|editing help can be found here]]''' (including adding pages, using mathematical formulas and embedding videos).

Temp VertexTransitive

2022-02-22T17:54:54Z

Grahame: /* Table of the orders of the largest known vertex-transitive graphs for the undirected degree diameter problem */

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

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

Below is the table of the largest known vertex-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: #f0f;" | '''10''' || style="background-color: #eee;" | '''14''' || style="background-color: #ff0;" | '''30''' ||style="background-color: #eee;" | '''60''' || style="background-color: #ff0;" | '''82''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #ff0;" | 546 ||style="background-color: #ff0;" | 1 250
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #eee;" | 30 ||style="background-color: #eee;" | 84 || style="background-color: #eee;" | 216 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | 18 ||style="background-color: #eee;" | 60 ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" | 32 ||style="background-color: #eee;" | 108 ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #0ff;" |'''50''' ||style="background-color: #eee;" |168 ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" | 48 ||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | 60 || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | 72 || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | 84 ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | 96 ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #f80;" | 162 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #f80;" | 338 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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;" | * || Cayley graphs; see the [[The_Degree_Diameter_Problem_for_Cayley_Graphs | separate page]] for details.
|-
|style="background-color: #f0f; text-align: center;" | * || The [https://mathworld.wolfram.com/PetersenGraph.html Petersen graph].
|-
|style="background-color: #ff0; text-align: center;" | * || See P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|-
|style="background-color: #0ff; text-align: center;" | * || The [https://mathworld.wolfram.com/Hoffman-SingletonGraph.html Hoffman-Singleton graph].
|-
|style="background-color: #f80; text-align: center;" | * || See B. D. McKay, M. Miller and J. Širáň, ''A note on large graphs of diameter two and given maximum degree''.
|}
</center>

Temp VertexTransitive

2022-02-20T16:20:45Z

Grahame: /* Table of the orders of the largest known vertex-transitive graphs for the undirected degree diameter problem */

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

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

Below is the table of the largest known vertex-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: #f0f;" | '''10''' || style="background-color: #eee;" | '''14''' || style="background-color: #ff0;" | '''30''' ||style="background-color: #eee;" | '''60''' || style="background-color: #ff0;" | '''82''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #ff0;" | 546 ||style="background-color: #ff0;" | 1 250
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #eee;" | '''30''' ||style="background-color: #eee;" | '''84''' || style="background-color: #eee;" | 216 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | '''18''' ||style="background-color: #eee;" | '''60''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" |'''32'''||style="background-color: #eee;" | '''108''' ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #0ff;" |'''50''' ||style="background-color: #eee;" |'''168''' ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" |'''48'''||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | '''72''' || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | '''84''' ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | '''96''' ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #f80;" | 162 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #f80;" | 338 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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;" | * || Cayley graphs; see the [[The_Degree_Diameter_Problem_for_Cayley_Graphs | separate page]] for details.
|-
|style="background-color: #f0f; text-align: center;" | * || The [https://mathworld.wolfram.com/PetersenGraph.html Petersen graph].
|-
|style="background-color: #ff0; text-align: center;" | * || See P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|-
|style="background-color: #0ff; text-align: center;" | * || The [https://mathworld.wolfram.com/Hoffman-SingletonGraph.html Hoffman-Singleton graph].
|-
|style="background-color: #f80; text-align: center;" | * || See B. D. McKay, M. Miller and J. Širáň, ''A note on large graphs of diameter two and given maximum degree''.
|}
</center>

Temp VertexTransitive

2022-02-19T18:38:45Z

Grahame:

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

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

Below is the table of the largest known vertex-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: #f0f;" | '''10''' || style="background-color: #eee;" | '''14''' || style="background-color: #ff0;" | '''30''' ||style="background-color: #eee;" | '''60''' || style="background-color: #ff0;" | '''82''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #ff0;" | 546 ||style="background-color: #ff0;" | 1 250
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #eee;" | '''30''' ||style="background-color: #eee;" | '''84''' || style="background-color: #eee;" | 216 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | '''18''' ||style="background-color: #eee;" | '''60''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" |'''32'''||style="background-color: #eee;" | '''108''' ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #eee;" |'''36''' ||style="background-color: #eee;" |'''168''' ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" |'''48'''||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | '''72''' || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | '''84''' ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | '''96''' ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #eee;" | 112 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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;" | * || Cayley graphs; see the [[The_Degree_Diameter_Problem_for_Cayley_Graphs | separate page]] for details.
|-
|style="background-color: #f0f; text-align: center;" | * || The [https://mathworld.wolfram.com/PetersenGraph.html Petersen graph].
|-
|style="background-color: #ff0; text-align: center;" | * || See P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|}
</center>

Temp VertexTransitive

2022-02-19T18:36:42Z

Grahame: /* Table of the orders of the largest known vertex-transitive graphs for the undirected degree diameter problem */

This is a temporary page for tables of vertex-transitive graphs in the degree-diameter problem.

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

Below is the table of the largest known vertex-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: #f0f;" | '''10''' || style="background-color: #eee;" | '''14''' || style="background-color: #ff0;" | '''30''' ||style="background-color: #eee;" | '''60''' || style="background-color: #ff0;" | '''82''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #ff0;" | 546 ||style="background-color: #ff0;" | 1 250
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #eee;" | '''30''' ||style="background-color: #eee;" | '''84''' || style="background-color: #eee;" | 216 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | '''18''' ||style="background-color: #eee;" | '''60''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" |'''32'''||style="background-color: #eee;" | '''108''' ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #eee;" |'''36''' ||style="background-color: #eee;" |'''168''' ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" |'''48'''||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | '''72''' || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | '''84''' ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | '''96''' ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #eee;" | 112 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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;" | * || Cayley graphs; see the [[The_Degree_Diameter_Problem_for_Cayley_Graphs | separate page]] for details.
|-
|style="background-color: #f0f; text-align: center;" | * || The [https://mathworld.wolfram.com/PetersenGraph.html Petersen graph].
|-
|style="background-color: #ff0; text-align: center;" | * || See P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|}
</center>

Temp VertexTransitive

2022-02-19T18:23:26Z

Grahame: /* Table of the orders of the largest known vertex-transitive graphs for the undirected degree diameter problem */

This is a temporary page for tables of vertex-transitive graphs in the degree-diameter problem.

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

Below is the table of the largest known vertex-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;" | '''8''' || style="background-color: #eee;" | '''14''' || style="background-color: #eee;" | '''24''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | '''72''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #eee;" | 506||style="background-color: #eee;" | 882
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #eee;" | '''30''' ||style="background-color: #eee;" | '''84''' || style="background-color: #eee;" | 216 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | '''18''' ||style="background-color: #eee;" | '''60''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" |'''32'''||style="background-color: #eee;" | '''108''' ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #eee;" |'''36''' ||style="background-color: #eee;" |'''168''' ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" |'''48'''||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | '''72''' || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | '''84''' ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | '''96''' ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #eee;" | 112 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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;" | * || Cayley graphs; see the [[The_Degree_Diameter_Problem_for_Cayley_Graphs | separate page]] for details.
|-
|style="background-color: #FF0066; text-align: center;" | * || Graphs found by Michael J. Dinneen and Paul Hafner. More details are available in a paper by the authors.
|-
|style="background-color: #993300; text-align: center;" | * || Graph found by Mitjana M. and Francesc Comellas. This graph was also found independently by Michael Sampels.
|-
|style="background-color: #CCCCFF; text-align: center;" | * || Graph found by Wohlmuth, and shown to be optimal by Marston Conder.
|-
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by Michael Sampels.
|-
|style="background-color: #ccff33; text-align: center;" | * || Graphs found (and verified as optimal in most cases) by Marston Conder. See [[Description of optimal Cayley graphs found by Marston Conder|Graphs found by Marston Conder]] for more details.
|-
|style="background-color: #339900; text-align: center;" | * || Optimal graph found by Marston Conder. This graph was also found independently by Eyal Loz.
|-
|style="background-color: #006600; text-align: center;" | * || Graph found by Eugene Curtin, and shown to be optimal by Marston Conder. This graph was also found independently by Eyal Loz.
|-
|style="background-color: #ff6600; text-align: center;" | * || Graphs found by Eyal Loz as part of the joint project ''The degree/diameter problem for several classes of graphs'' by E. Loz, H. Pérez-Rosés and G. Pineda-Villavicencio.
|-
|style="background-color: #FF9900; text-align: center;" | * || Graphs found by Eyal Loz. More details are available in a paper by Eyal Loz and Jozef Širáň.
|-
|style="background-color: #ffff66; text-align: center;" | * || Graphs found by Eyal Loz and Guillermo Pineda-Villavicencio. More details are available in a paper by the authors.
|-
|style="background-color: #ff9999; text-align: center;" | * || Graph found by P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by Marcel Abas.
|}
</center>

Temp VertexTransitive

2022-02-19T18:21:25Z

Grahame: /* Table of the orders of the largest known vertex-transitive graphs for the undirected degree diameter problem */

This is a temporary page for tables of vertex-transitive graphs in the degree-diameter problem.

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

Below is the table of the largest known vertex-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;" | '''8''' || style="background-color: #eee;" | '''14''' || style="background-color: #eee;" | '''24''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | '''72''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #eee;" | 506||style="background-color: #eee;" | 882
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #eee;" | '''30''' ||style="background-color: #eee;" | '''84''' || style="background-color: #eee;" | 216 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | '''18''' ||style="background-color: #eee;" | '''60''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" |'''32'''||style="background-color: #eee;" | '''108''' ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #eee;" |'''36''' ||style="background-color: #eee;" |'''168''' ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" |'''48'''||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | '''72''' || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | '''84''' ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | '''96''' ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #eee;" | 112 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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;" | * || Cayley graphs; see the page for details.
|-
|style="background-color: #FF0066; text-align: center;" | * || Graphs found by Michael J. Dinneen and Paul Hafner. More details are available in a paper by the authors.
|-
|style="background-color: #993300; text-align: center;" | * || Graph found by Mitjana M. and Francesc Comellas. This graph was also found independently by Michael Sampels.
|-
|style="background-color: #CCCCFF; text-align: center;" | * || Graph found by Wohlmuth, and shown to be optimal by Marston Conder.
|-
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by Michael Sampels.
|-
|style="background-color: #ccff33; text-align: center;" | * || Graphs found (and verified as optimal in most cases) by Marston Conder. See [[Description of optimal Cayley graphs found by Marston Conder|Graphs found by Marston Conder]] for more details.
|-
|style="background-color: #339900; text-align: center;" | * || Optimal graph found by Marston Conder. This graph was also found independently by Eyal Loz.
|-
|style="background-color: #006600; text-align: center;" | * || Graph found by Eugene Curtin, and shown to be optimal by Marston Conder. This graph was also found independently by Eyal Loz.
|-
|style="background-color: #ff6600; text-align: center;" | * || Graphs found by Eyal Loz as part of the joint project ''The degree/diameter problem for several classes of graphs'' by E. Loz, H. Pérez-Rosés and G. Pineda-Villavicencio.
|-
|style="background-color: #FF9900; text-align: center;" | * || Graphs found by Eyal Loz. More details are available in a paper by Eyal Loz and Jozef Širáň.
|-
|style="background-color: #ffff66; text-align: center;" | * || Graphs found by Eyal Loz and Guillermo Pineda-Villavicencio. More details are available in a paper by the authors.
|-
|style="background-color: #ff9999; text-align: center;" | * || Graph found by P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by Marcel Abas.
|}
</center>

Temp VertexTransitive

2022-02-19T18:18:15Z

Grahame: /* Table of the orders of the largest known vertex-transitive graphs for the undirected degree diameter problem */

This is a temporary page for tables of vertex-transitive graphs in the degree-diameter problem.

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

Below is the table of the largest known vertex-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''' || '''8''' || style="background-color: #eee;" | '''14''' || style="background-color: #eee;" | '''24''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | '''72''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #eee;" | 506||style="background-color: #eee;" | 882
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #eee;" | '''30''' ||style="background-color: #eee;" | '''84''' || style="background-color: #eee;" | 216 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | '''18''' ||style="background-color: #eee;" | '''60''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" |'''32'''||style="background-color: #eee;" | '''108''' ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #eee;" |'''36''' ||style="background-color: #eee;" |'''168''' ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" |'''48'''||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | '''72''' || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | '''84''' ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | '''96''' ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #eee;" | 112 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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;" | * || Cayley graphs; see the page for details.
|-
|style="background-color: #FF0066; text-align: center;" | * || Graphs found by Michael J. Dinneen and Paul Hafner. More details are available in a paper by the authors.
|-
|style="background-color: #993300; text-align: center;" | * || Graph found by Mitjana M. and Francesc Comellas. This graph was also found independently by Michael Sampels.
|-
|style="background-color: #CCCCFF; text-align: center;" | * || Graph found by Wohlmuth, and shown to be optimal by Marston Conder.
|-
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by Michael Sampels.
|-
|style="background-color: #ccff33; text-align: center;" | * || Graphs found (and verified as optimal in most cases) by Marston Conder. See [[Description of optimal Cayley graphs found by Marston Conder|Graphs found by Marston Conder]] for more details.
|-
|style="background-color: #339900; text-align: center;" | * || Optimal graph found by Marston Conder. This graph was also found independently by Eyal Loz.
|-
|style="background-color: #006600; text-align: center;" | * || Graph found by Eugene Curtin, and shown to be optimal by Marston Conder. This graph was also found independently by Eyal Loz.
|-
|style="background-color: #ff6600; text-align: center;" | * || Graphs found by Eyal Loz as part of the joint project ''The degree/diameter problem for several classes of graphs'' by E. Loz, H. Pérez-Rosés and G. Pineda-Villavicencio.
|-
|style="background-color: #FF9900; text-align: center;" | * || Graphs found by Eyal Loz. More details are available in a paper by Eyal Loz and Jozef Širáň.
|-
|style="background-color: #ffff66; text-align: center;" | * || Graphs found by Eyal Loz and Guillermo Pineda-Villavicencio. More details are available in a paper by the authors.
|-
|style="background-color: #ff9999; text-align: center;" | * || Graph found by P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by Marcel Abas.
|}
</center>

Temp VertexTransitive

2022-02-19T18:15:35Z

Grahame:

This is a temporary page for tables of vertex-transitive graphs in the degree-diameter problem.

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

Below is the table of the largest known vertex-transitive graph 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''' || '''8''' || style="background-color: #eee;" | '''14''' || style="background-color: #eee;" | '''24''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | '''72''' ||style="background-color: #eee;" | '''168''' ||style="background-color: #eee;" | '''300'''||style="background-color: #eee;" | 506||style="background-color: #eee;" | 882
|-

| '''4''' ||style="background-color: #eee;" | '''13''' ||style="background-color: #eee;" | '''30''' ||style="background-color: #eee;" | '''84''' || style="background-color: #eee;" | 216 ||style="background-color: #eee;" | 513||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 3 080 ||style="background-color: #eee;" | 7 550 ||style="background-color: #eee;" | 17 604
|-
| '''5''' ||style="background-color: #eee;" | '''18''' ||style="background-color: #eee;" | '''60''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 546||style="background-color: #eee;" | 1 640 ||style="background-color: #eee;" | 5 500 ||style="background-color: #eee;" | 16 965 ||style="background-color: #eee;" | 57 840 ||style="background-color: #eee;" | 187 056
|-
| '''6''' || style="background-color: #eee;" |'''32'''||style="background-color: #eee;" | '''108''' ||style="background-color: #eee;" | 375 ||style="background-color: #eee;" | 1 395 ||style="background-color: #eee;" | 5 115 ||style="background-color: #eee;" | 19 383 ||style="background-color: #eee;" | 76 461 ||style="background-color: #eee;" | 307 845 ||style="background-color: #eee;" | 1 253 615
|-
| '''7''' ||style="background-color: #eee;" |'''36''' ||style="background-color: #eee;" |'''168''' ||style="background-color: #eee;" | 672 ||style="background-color: #eee;" | 2 756 ||style="background-color: #eee;" | 11 988 ||style="background-color: #eee;" | 52 768 ||style="background-color: #eee;" | 249 660 ||style="background-color: #eee;" | 1 223 050 ||style="background-color: #eee;" | 6 007 230
|-
| '''8''' ||style="background-color: #eee;" |'''48'''||style="background-color: #eee;" | 253 ||style="background-color: #eee;" | 1 100 ||style="background-color: #eee;" | 5 060 ||style="background-color: #eee;" | 23 991 ||style="background-color: #eee;" | 131 137 ||style="background-color: #eee;" | 734 820 ||style="background-color: #eee;" | 4 243 100 ||style="background-color: #eee;" | 24 897 161
|-
| '''9''' ||style="background-color: #eee;" | '''60''' || style="background-color: #eee;" | 294 ||style="background-color: #eee;" | 1 550 ||style="background-color: #eee;" | 8 200 ||style="background-color: #eee;" | 45 612 ||style="background-color: #eee;" | 279 616 ||style="background-color: #eee;" | 1 686 600 ||style="background-color: #eee;" | 12 123 288 ||style="background-color: #eee;" | 65 866 350
|-
| '''10''' ||style="background-color: #eee;" | '''72''' || style="background-color: #eee;" | 406 ||style="background-color: #eee;" | 2 286 ||style="background-color: #eee;" | 13 140 ||style="background-color: #eee;" | 81 235 ||style="background-color: #eee;" | 583 083 ||style="background-color: #eee;" | 4 293 452 ||style="background-color: #eee;" | 27 997 191 ||style="background-color: #eee;" | 201 038 922
|-
| '''11''' ||style="background-color: #eee;" | '''84''' ||style="background-color: #eee;" | 486 || style="background-color: #eee;" | 2 860 ||style="background-color: #eee;" | 19 500 ||style="background-color: #eee;" | 139 446 ||style="background-color: #eee;" | 1 001 268 ||style="background-color: #eee;" | 7 442 328 || style="background-color: #eee;" | 72 933 102 ||style="background-color: #eee;" | 500 605 110
|-
| '''12''' ||style="background-color: #eee;" | '''96''' ||style="background-color: #eee;" | 605 ||style="background-color: #eee;" | 3 775 ||style="background-color: #eee;" | 29 470||style="background-color: #eee;" | 229 087 ||style="background-color: #eee;" | 1 999 500 ||style="background-color: #eee;" | 15 924 326 ||style="background-color: #eee;" | 158 158 875 ||style="background-color: #eee;" | 1 225 374 192
|-
| '''13''' ||style="background-color: #eee;" | 112 ||style="background-color: #eee;" | 680 ||style="background-color: #eee;" | 4 788 ||style="background-color: #eee;" |40 260 ||style="background-color: #eee;" | 347 126 ||style="background-color: #eee;" | 3 322 080 ||style="background-color: #eee;" | 29 927 790 ||style="background-color: #eee;" | 233 660 788 ||style="background-color: #eee;" | 2 129 329 324
|-
| '''14''' ||style="background-color: #eee;" | 128 ||style="background-color: #eee;" | 873 ||style="background-color: #eee;" | 6 510 ||style="background-color: #eee;" | 57 837 ||style="background-color: #eee;" | 530 448 ||style="background-color: #eee;" | 5 600 532 ||style="background-color: #eee;" | 50 128 239 ||style="background-color: #eee;" | 579 328 377 ||style="background-color: #eee;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #eee;" | 144 || style="background-color: #eee;" | 972 || style="background-color: #eee;" | 7 956 ||style="background-color: #eee;" | 76 518 || style="background-color: #eee;" | 787 116 ||style="background-color: #eee;" | 8 599 986 ||style="background-color: #eee;" | 88 256 520 ||style="background-color: #eee;" | 1 005 263 436 ||style="background-color: #eee;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 155 ||style="background-color: #eee;" | 9 576 ||style="background-color: #eee;" | 100 650 || style="background-color: #eee;" | 1 125 264 || style="background-color: #eee;" | 12 500 082 ||style="background-color: #eee;" | 135 340 551 ||style="background-color: #eee;" | 1 995 790 371 ||style="background-color: #eee;" | 12 951 451 931
|-
| '''17''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 260 ||style="background-color: #eee;" | 12 090 ||style="background-color: #eee;" | 133 144 || style="background-color: #eee;" | 1 609 830 || style="background-color: #eee;" | 18 495 162 ||style="background-color: #eee;" | 220 990 700 ||style="background-color: #eee;" | 3 372 648 954 ||style="background-color: #eee;" |

|-
| '''18''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 510 ||style="background-color: #eee;" | 15 026 ||style="background-color: #eee;" | 171 828 || style="background-color: #eee;" | 2 193 321 || style="background-color: #eee;" | 26 515 120 ||style="background-color: #eee;" | 323 037 476 ||style="background-color: #eee;" | 5 768 971 167 ||style="background-color: #eee;" |

|-
| '''19''' ||style="background-color: #eee;" | 200 ||style="background-color: #eee;" | 1 638 ||style="background-color: #eee;" | 17 658 ||style="background-color: #eee;" | 221 676 || style="background-color: #eee;" | 3 030 544 || style="background-color: #eee;" | 39 123 116 ||style="background-color: #eee;" | 501 001 000 ||style="background-color: #eee;" | 8 855 580 344 ||style="background-color: #eee;" |

|-
| '''20''' ||style="background-color: #eee;" | 210 ||style="background-color: #eee;" | 1 958 ||style="background-color: #eee;" | 21 333 ||style="background-color: #eee;" | 281 820 || style="background-color: #eee;" | 4 040 218 || style="background-color: #eee;" | 55 625 185 ||style="background-color: #eee;" | 762 374 779 ||style="background-color: #eee;" | 12 951 451 931 ||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: #FF0066; text-align: center;" | * || Graphs found by Michael J. Dinneen and Paul Hafner. More details are available in a paper by the authors.
|-
|style="background-color: #993300; text-align: center;" | * || Graph found by Mitjana M. and Francesc Comellas. This graph was also found independently by Michael Sampels.
|-
|style="background-color: #CCCCFF; text-align: center;" | * || Graph found by Wohlmuth, and shown to be optimal by Marston Conder.
|-
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by Michael Sampels.
|-
|style="background-color: #ccff33; text-align: center;" | * || Graphs found (and verified as optimal in most cases) by Marston Conder. See [[Description of optimal Cayley graphs found by Marston Conder|Graphs found by Marston Conder]] for more details.
|-
|style="background-color: #339900; text-align: center;" | * || Optimal graph found by Marston Conder. This graph was also found independently by Eyal Loz.
|-
|style="background-color: #006600; text-align: center;" | * || Graph found by Eugene Curtin, and shown to be optimal by Marston Conder. This graph was also found independently by Eyal Loz.
|-
|style="background-color: #ff6600; text-align: center;" | * || Graphs found by Eyal Loz as part of the joint project ''The degree/diameter problem for several classes of graphs'' by E. Loz, H. Pérez-Rosés and G. Pineda-Villavicencio.
|-
|style="background-color: #FF9900; text-align: center;" | * || Graphs found by Eyal Loz. More details are available in a paper by Eyal Loz and Jozef Širáň.
|-
|style="background-color: #ffff66; text-align: center;" | * || Graphs found by Eyal Loz and Guillermo Pineda-Villavicencio. More details are available in a paper by the authors.
|-
|style="background-color: #ff9999; text-align: center;" | * || Graph found by P. Potočnik, P. Spiga and G. Verret, ''Cubic vertex-transitive graphs on up to 1280 vertices''.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by Marcel Abas.
|}
</center>

Temp VertexTransitive

2022-02-18T11:04:43Z

Grahame: Created page with "This is a temporary page for tables of vertex-transitive graphs in the degree-diameter problem."

This is a temporary page for tables of vertex-transitive graphs in the degree-diameter problem.

Tables and Results

2022-01-21T15:19:23Z

Grahame: /* Table 2: bounds for trivalent cages */

==Trivalent cages==
===Table 1: known trivalent cages===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''g'' ''' || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''|| style="background-color: #cccccc;" | '''8'''|| style="background-color: #cccccc;" | '''9'''|| style="background-color: #cccccc;" | '''10'''|| style="background-color: #cccccc;" | '''11'''|| style="background-color: #cccccc;" | '''12'''
|-
|style="background-color: #cccccc;"| ''' ''n(3,g)'' ''' || 10 ||14 ||24 ||30 ||58 ||70 ||112 ||126
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 1|| 1|| 18|| 3|| 1|| 1
|}
</center>

===Table 2: bounds for trivalent cages===
Optimal graphs are marked in bold
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Girth ''g'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' ||style="background-color: #cccccc;"| ''' Number of cages''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"| 5 ||style="background-color: #cccccc;"| 10 || '''10''' ||1 ||[http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"| 6 ||style="background-color: #cccccc;"| 14 ||'''14'''|| 1 ||[http://en.wikipedia.org/wiki/Heawood_graph Heawood]
|-
|style="background-color: #cccccc;"| 7 ||style="background-color: #cccccc;"| 24 || '''24''' ||1 ||McGee
|-
|style="background-color: #cccccc;"| 8 ||style="background-color: #cccccc;"| 30 ||'''30'''|| 1 ||[http://en.wikipedia.org/wiki/Tutte_eight_cage Tutte]
|-
|style="background-color: #cccccc;"| 9 ||style="background-color: #cccccc;"| 58 ||'''58''' ||18 ||Brinkmann-McKay-Saager
|-
|style="background-color: #cccccc;"| 10 ||style="background-color: #cccccc;"| 70 ||'''70''' ||3 ||O’Keefe-Wong
|-
|style="background-color: #cccccc;"| 11 ||style="background-color: #cccccc;"| 112 || '''112'''|| 1 ||McKay-Myrvold; Balaban
|-
|style="background-color: #cccccc;"| 12 ||style="background-color: #cccccc;"| 126 || '''126'''|| 1|| Benson
|-
|style="background-color: #cccccc;"| 13 ||style="background-color: #cccccc;"| 202 ||272|| ||McKay-Myrvold; Hoare
|-
|style="background-color: #cccccc;"| 14 ||style="background-color: #cccccc;"| 258 ||384 ||||McKay; Exoo
|-
|style="background-color: #cccccc;"| 15 ||style="background-color: #cccccc;"| 384 ||620|||| Biggs
|-
|style="background-color: #cccccc;"| 16 ||style="background-color: #cccccc;"| 512 ||960|| ||Exoo
|-
|style="background-color: #cccccc;"| 17 ||style="background-color: #cccccc;"| 768 ||2176|| ||Exoo
|-
|style="background-color: #cccccc;"| 18 ||style="background-color: #cccccc;"| 1024 ||2560|| ||Exoo
|-
|style="background-color: #cccccc;"| 19 ||style="background-color: #cccccc;"| 1536 ||4324|||| Hoare, H(47)
|-
|style="background-color: #cccccc;"| 20 ||style="background-color: #cccccc;"| 2048 || 5376 || ||Exoo
|-
|style="background-color: #cccccc;"| 21 ||style="background-color: #cccccc;"| 3072 ||16028 || ||Exoo
|-
|style="background-color: #cccccc;"| 22 ||style="background-color: #cccccc;"| 4096 || 16206|| || Biggs-Hoare, S(73)
|-
|style="background-color: #cccccc;"| 23 ||style="background-color: #cccccc;"| 6144 ||35446 || ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 24 ||style="background-color: #cccccc;"| 8192 || 35640 || ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 25 ||style="background-color: #cccccc;"| 12288 ||108906|| || Exoo
|-
|style="background-color: #cccccc;"| 26 ||style="background-color: #cccccc;"| 16384 || 109200 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 27 ||style="background-color: #cccccc;"| 24576 ||285852 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 28 ||style="background-color: #cccccc;"| 32768 ||368640|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 29 ||style="background-color: #cccccc;"| 49152 ||805746|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 30 ||style="background-color: #cccccc;"| 65536 ||806736|| || Erskine-Tuite
|-
|style="background-color: #cccccc;"| 31 ||style="background-color: #cccccc;"| 98304 ||1440338|| ||Erskine-Tuite
|-
|style="background-color: #cccccc;"| 32 ||style="background-color: #cccccc;"| 131072 ||1441440|| || Erskine-Tuite
|}
</center>

==Cages of girth 5 and 6==
Optimal graphs are marked in bold.
===Table 1: known cages of Girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''k'' ''' || style="background-color: #cccccc;" | 3 || style="background-color: #cccccc;" | 4 || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''
|-
|style="background-color: #cccccc;"| ''' ''n(k,5)'' ''' || 10 || 19|| 30|| 40|| 50
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 4|| 1|| 1
|}
</center>

===Table 2: bounds for cages of girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"|10 ||'''10'''|| [http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|19 ||'''19'''|| Robertson
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|30 ||'''30'''|| Robertson-Wegner-Wong
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|40 ||'''40'''|| Wong
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|50||''' 50'''|| [http://en.wikipedia.org/wiki/Hoffman-Singleton_graph Hoﬀman-Singleton]
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|67|| 80|| Royle
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|86|| 96 ||Jørgensen
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|103|| 126 ||Exoo
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|124 ||156|| Jørgensen
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|147|| 203|| Exoo
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|174 ||240 ||Exoo
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|199 ||288 ||Jørgensen
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|230 ||312 ||Jørgensen
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|259 ||336|| Jørgensen
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|294|| 448 ||Schwenk
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|327|| 480|| Schwenk
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|364|| 512|| Schwenk
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|403|| 576 ||Jørgensen
|}
</center>

===Table 3: bounds for cages of girth 6===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"| 14 ||'''14'''|| [http://en.wikipedia.org/wiki/Projective_plane Projective Plane]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|26 ||'''26'''|| Projective Plane
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|42 ||'''42'''|| Projective Plane
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|62 ||'''62'''|| Projective Plane
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|90 ||'''90'''|| O’Keefe-Wong
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|114 ||'''114'''|| Projective Plane
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|146 ||'''146'''|| Projective Plane
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|182 ||'''182'''|| Projective Plane
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|224 ||240|| Wong
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|266|| 266|| Projective Plane
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|314 ||336|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|366 ||'''366'''|| Projective Plane
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|422 ||462|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|482 ||504|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|546 ||'''546'''|| Projective Plane
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|614 ||'''614'''|| Projective Plane
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|686 ||720 ||Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|762 ||'''762'''|| Projective Plane
|}
</center>

[[Category:The Cage Problem]]

Mirka Miller's Combinatorics Webinar Series

2021-04-07T09:25:56Z

Grahame: /* Upcoming Talks */

Welcome to the home page for Mirka Miller's Combinatorics Webinar Series.

[[File:Mirka2.jpg]]

==Upcoming Talks==

'''Date: Wednesday April 14 2021'''

'''Time: 1000 Bratislava (0900 UK)'''

'''Speaker: Prof. Jozef Širáň'''

'''Title: May there be many more repeats'''

'''Meeting link:''' [https://meet.google.com/kgc-uwpc-ngp https://meet.google.com/kgc-uwpc-ngp]

'''Abstract:''' This is my reminiscence on two mathematical aspects of my collaboration with Mirka Miller in the degree-diameter problem: the lifting technique in constructions of `large' examples and her method of repeats in non-existence proofs.

==Previous Talks==

Mirka Miller's Combinatorics Webinar Series

2021-04-07T09:24:18Z

Grahame: /* Upcoming Talks */

Welcome to the home page for Mirka Miller's Combinatorics Webinar Series.

[[File:Mirka2.jpg]]

==Upcoming Talks==

'''Wednesday April 14 2021 1000 Bratislava (0900 UK)'''

'''Prof. Jozef Širáň'''

'''Title: May there be many more repeats''' [https://meet.google.com/kgc-uwpc-ngp https://meet.google.com/kgc-uwpc-ngp]

'''Abstract:''' This is my reminiscence on two mathematical aspects of my collaboration with Mirka Miller in the degree-diameter problem: the lifting technique in constructions of `large' examples and her method of repeats in non-existence proofs.

==Previous Talks==

Mirka Miller's Combinatorics Webinar Series

2021-04-06T09:56:45Z

Grahame: /* Upcoming Talks */

Welcome to the home page for Mirka Miller's Combinatorics Webinar Series.

[[File:Mirka2.jpg]]

==Upcoming Talks==

Wednesday April 14 2021 (time TBC)

Prof. Jozef Širáň

Title TBC [https://meet.google.com/kgc-uwpc-ngp https://meet.google.com/kgc-uwpc-ngp]

==Previous Talks==

Mirka Miller's Combinatorics Webinar Series

2021-04-06T09:53:04Z

Grahame: /* Upcoming Talks */

Mirka Miller's Combinatorics Webinar Series

2021-04-06T09:50:29Z

Grahame:

Welcome to the home page for Mirka Miller's Combinatorics Webinar Series.

[[File:Mirka2.jpg]]

==Upcoming Talks==

==Previous Talks==

File:Mirka2.jpg

2021-04-06T09:48:49Z

Grahame:

Mirka Miller's Combinatorics Webinar Series

2021-04-06T09:48:23Z

Grahame:

Welcome to the home page for Mirka Miller's Combinatorics Webinar Series.

[[File:mirka2.jpg]]

==Upcoming Talks==

==Previous Talks==

Main Page

2021-04-06T09:44:53Z

Grahame: /* Meetings, seminars and talks */

==About Combinatorics Wiki==

Combinatorics Wiki is a wiki presenting the latest results on problems in various topics in the field of [http://en.wikipedia.org/wiki/Combinatorics combinatorics]. Combinatorics Wiki will only allow updates by active expert researchers in their fields, with the following goals:

* Creating a stable venue for researchers to announce published and pre-published work in real time. As many of the existing problems, in particular in extremal theory are of highly competitive nature, where new results very often supersede existing results, an up to date resource listing the most current results is therefore essential to the community working in a specific field. Taking into account the long time it can take to publish mathematical papers, it can be very helpful to announce and briefly describe new findings before the actual publication.

* Creating an extensive peer-reviewed source of information, allowing for new and existing researchers to stay up to date with work done by others in their field.

* Keeping a detailed history of previous work, findings, publications and results, in a simple user friendly wiki format.

* Allowing registered users to review and comment on unpublished and published work by other users.

* Giving supervisors and students ideas for new projects and open problems.

* Creating a stable community of researchers in different areas, and promoting collaborations.

==List of problem areas==

* [[Enumeration of Latin Squares and Rectangles]]

* [[The Cage Problem|The Cage Problem or The Degree/Girth Problem]]

* [[The Degree/Diameter Problem]]

* [[The Maximum Degree-and-Diameter-Bounded Subgraph Problem]]

* [[Minor-Closed Classes of Matroids]]

* [[Ramsey Theory]]

* [[Extremal C_t-free graphs]]

==List of video channels==

* [[Lectures Available Online|Lectures available online]]

* [[Documentaries Available Online|Documentaries available online]]

==Supporting organizations==

* [http://combinatorics-australasia.org/ CMSA - Combinatorial Mathematics Society of Australasia]

* [http://graphtheorygroup.com GTA - Graph Theory and Applications - The University of Newcastle, Australia]

* [http://www.indstate.edu/home.htm Indiana State University]

* [http://www-mat.upc.es/grup_de_grafs Research Group on Graph Theory and Combinatorics - UPC, Spain]

==Newsletters==

[[Combinatorial_Mathematics_Society_of_Australasia_Newsletters|Newsletters of the Combinatorial Mathematics Society of Australasia]]

==Meetings, seminars and talks==

Links to upcoming talks which may be of interest to researchers in the problem areas covered by Combinatorics Wiki.

[[Mirka_Miller%27s_Combinatorics_Webinar_Series|Mirka Miller's Combinatorics Webinar Series]]

==Employment Opportunities==

Members of the Combinatorics Wiki community are welcome to [[Employment Opportunities| advertise research internships, postdoc positions and other research and teaching openings]] in their respective institutions and others.

==Combinatorics Wiki rules==

Please read our rules of [[Rules and Regulations|use of Combinatorics Wiki]].

==Note for potential new contributors and moderators==

We are always interested in extending our list of problem areas. Please contact our [[List of Moderators|moderators]] with new ideas and suggestions. New registered users, '''[[Help:Editing|editing help can be found here]]''' (including adding pages, using mathematical formulas and embedding videos).

Mirka Miller's Combinatorics Webinar Series

2021-04-06T09:43:49Z

Grahame: Grahame moved page Mirka Miller Combinatorics Webinar to Mirka Miller's Combinatorics Webinar Series

==Talks==

Mirka Miller Combinatorics Webinar

2021-04-06T09:43:49Z

Grahame: Grahame moved page Mirka Miller Combinatorics Webinar to Mirka Miller's Combinatorics Webinar Series

#REDIRECT [[Mirka Miller's Combinatorics Webinar Series]]

Mirka Miller's Combinatorics Webinar Series

2021-04-06T09:43:22Z

Grahame: Created page with "==Talks=="

==Talks==

Main Page

2021-04-06T09:41:43Z

Grahame: /* Meetings, seminars and talks */

==About Combinatorics Wiki==

Combinatorics Wiki is a wiki presenting the latest results on problems in various topics in the field of [http://en.wikipedia.org/wiki/Combinatorics combinatorics]. Combinatorics Wiki will only allow updates by active expert researchers in their fields, with the following goals:

* Creating a stable venue for researchers to announce published and pre-published work in real time. As many of the existing problems, in particular in extremal theory are of highly competitive nature, where new results very often supersede existing results, an up to date resource listing the most current results is therefore essential to the community working in a specific field. Taking into account the long time it can take to publish mathematical papers, it can be very helpful to announce and briefly describe new findings before the actual publication.

* Creating an extensive peer-reviewed source of information, allowing for new and existing researchers to stay up to date with work done by others in their field.

* Keeping a detailed history of previous work, findings, publications and results, in a simple user friendly wiki format.

* Allowing registered users to review and comment on unpublished and published work by other users.

* Giving supervisors and students ideas for new projects and open problems.

* Creating a stable community of researchers in different areas, and promoting collaborations.

==List of problem areas==

* [[Enumeration of Latin Squares and Rectangles]]

* [[The Cage Problem|The Cage Problem or The Degree/Girth Problem]]

* [[The Degree/Diameter Problem]]

* [[The Maximum Degree-and-Diameter-Bounded Subgraph Problem]]

* [[Minor-Closed Classes of Matroids]]

* [[Ramsey Theory]]

* [[Extremal C_t-free graphs]]

==List of video channels==

* [[Lectures Available Online|Lectures available online]]

* [[Documentaries Available Online|Documentaries available online]]

==Supporting organizations==

* [http://combinatorics-australasia.org/ CMSA - Combinatorial Mathematics Society of Australasia]

* [http://graphtheorygroup.com GTA - Graph Theory and Applications - The University of Newcastle, Australia]

* [http://www.indstate.edu/home.htm Indiana State University]

* [http://www-mat.upc.es/grup_de_grafs Research Group on Graph Theory and Combinatorics - UPC, Spain]

==Newsletters==

[[Combinatorial_Mathematics_Society_of_Australasia_Newsletters|Newsletters of the Combinatorial Mathematics Society of Australasia]]

==Meetings, seminars and talks==

Links to upcoming talks which may be of interest to researchers in the problem areas covered by Combinatorics Wiki.

[[Mirka_Miller_Combinatorics_Webinar|Mirka Miller's Combinatorics Webinar Series]]

==Employment Opportunities==

Members of the Combinatorics Wiki community are welcome to [[Employment Opportunities| advertise research internships, postdoc positions and other research and teaching openings]] in their respective institutions and others.

==Combinatorics Wiki rules==

Please read our rules of [[Rules and Regulations|use of Combinatorics Wiki]].

==Note for potential new contributors and moderators==

We are always interested in extending our list of problem areas. Please contact our [[List of Moderators|moderators]] with new ideas and suggestions. New registered users, '''[[Help:Editing|editing help can be found here]]''' (including adding pages, using mathematical formulas and embedding videos).

Main Page

2021-04-05T16:49:08Z

Grahame:

==About Combinatorics Wiki==

Combinatorics Wiki is a wiki presenting the latest results on problems in various topics in the field of [http://en.wikipedia.org/wiki/Combinatorics combinatorics]. Combinatorics Wiki will only allow updates by active expert researchers in their fields, with the following goals:

* Creating a stable venue for researchers to announce published and pre-published work in real time. As many of the existing problems, in particular in extremal theory are of highly competitive nature, where new results very often supersede existing results, an up to date resource listing the most current results is therefore essential to the community working in a specific field. Taking into account the long time it can take to publish mathematical papers, it can be very helpful to announce and briefly describe new findings before the actual publication.

* Creating an extensive peer-reviewed source of information, allowing for new and existing researchers to stay up to date with work done by others in their field.

* Keeping a detailed history of previous work, findings, publications and results, in a simple user friendly wiki format.

* Allowing registered users to review and comment on unpublished and published work by other users.

* Giving supervisors and students ideas for new projects and open problems.

* Creating a stable community of researchers in different areas, and promoting collaborations.

==List of problem areas==

* [[Enumeration of Latin Squares and Rectangles]]

* [[The Cage Problem|The Cage Problem or The Degree/Girth Problem]]

* [[The Degree/Diameter Problem]]

* [[The Maximum Degree-and-Diameter-Bounded Subgraph Problem]]

* [[Minor-Closed Classes of Matroids]]

* [[Ramsey Theory]]

* [[Extremal C_t-free graphs]]

==List of video channels==

* [[Lectures Available Online|Lectures available online]]

* [[Documentaries Available Online|Documentaries available online]]

==Supporting organizations==

* [http://combinatorics-australasia.org/ CMSA - Combinatorial Mathematics Society of Australasia]

* [http://graphtheorygroup.com GTA - Graph Theory and Applications - The University of Newcastle, Australia]

* [http://www.indstate.edu/home.htm Indiana State University]

* [http://www-mat.upc.es/grup_de_grafs Research Group on Graph Theory and Combinatorics - UPC, Spain]

==Newsletters==

[[Combinatorial_Mathematics_Society_of_Australasia_Newsletters|Newsletters of the Combinatorial Mathematics Society of Australasia]]

==Meetings, seminars and talks==

Links to upcoming talks which may be of interest to researchers in the problem areas covered by Combinatorics Wiki.

==Employment Opportunities==

Members of the Combinatorics Wiki community are welcome to [[Employment Opportunities| advertise research internships, postdoc positions and other research and teaching openings]] in their respective institutions and others.

==Combinatorics Wiki rules==

Please read our rules of [[Rules and Regulations|use of Combinatorics Wiki]].

==Note for potential new contributors and moderators==

We are always interested in extending our list of problem areas. Please contact our [[List of Moderators|moderators]] with new ideas and suggestions. New registered users, '''[[Help:Editing|editing help can be found here]]''' (including adding pages, using mathematical formulas and embedding videos).

The Degree Diameter Problem for Bipartite Graphs

2021-03-30T14:34:56Z

Grahame: /* Table of the orders of the largest known bipartite graphs */

==Introduction==
The '''degree/diameter problem for bipartite graphs''' can be stated as follows:

''Given natural numbers ''d'' and ''k'', find the largest possible number ''Nb(d,k)'' of vertices in a bipartite graph of maximum degree ''d'' and diameter ''k''.''

An upper bound for ''Nb(d,k)'' is given by the so-called ''bipartite Moore bound'' ''Mb(d,k)=2((d-1)k-2)(d-2)-1''. Bipartite ''(d,k)''-graphs whose order attains the bipartite Moore bound are called ''bipartite Moore graphs''.

Bipartite Moore graphs have proved to be very rare. Feit and Higman, and also independently Singleton, proved that such graphs exist only when the diameter is 2,3,4 or 6. In the cases when the diameter is 3, 4 or 6, they have been constructed only when ''d-1'' is a prime power.

Therefore, in attempting to settle the values of ''Nb(d,k)'', research activities in this problem have follow the following two directions:

*Increasing the lower bounds for ''Nb(d,k)'' by constructing ever larger graphs.

* Lowering and/or setting upper bounds for ''Nb(d,k)'' by proving the non-existence of graphs
whose order is close to the bipartite Moore bounds ''Mb(d,k)=2((d-1)k-1)(d-2)-1''.

==Increasing the lower bounds for ''Nb(d,k)''==
In recent years there has not been much activity in the constructions of large bipartite graphs. This may be, in part, because there was not an online table showing the latest constructions. In this direction Charles Delorme (in some cases collaborating with Bond and Gómez-Martí) provided some large bipartite graphs by using graph compounding, the concept of partial Cayley graph, and other techniques.

Now, with the release of this online table (see below), we expect to stimulate further research on this area.

Below is the table of the largest known [http://en.wikipedia.org/wiki/Bipartite_graph bipartite] graphs (as of January 2012) in the undirected [[The Degree/Diameter Problem | degree diameter problem]] for bipartite graphs of [http://en.wikipedia.org/wiki/Degree_(graph_theory) degree] at most 3 ≤ ''d'' ≤ 16 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 3 ≤ ''k'' ≤ 10. This table represents the best lower bounds known at present on the order of ''(d,k)''-bipartite graphs. Many of the graphs of diameter 3 ,4 and 6 are bipartite Moore graphs, and thus are optimal. All optimal graphs are marked in bold.

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

<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
| '''<math>d</math>\<math>k</math>'''|| '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10'''
|-
| '''3''' ||style="background-color: #bbffff;" | '''14'''||style="background-color: #bbffff;" | '''30''' ||style="background-color: #0066CC;" | '''56''' ||style="background-color: #bbffff;" | '''126''' ||style="background-color: #993300;" | 168||style="background-color: #66ff66;" | 256||style="background-color: #FF9900;" | 506 || style="background-color: #66ff66;" | 800
|-
| '''4''' || style="background-color: #bbffff;" | '''26''' ||style="background-color: #bbffff;" | '''80''' || style="background-color: #ff0000;" | 160 ||style="background-color: #bbffff;" | '''728''' ||style="background-color: #ff0000;" | 840 ||style="background-color: #FF9900;" | 2 184 ||style="background-color: #FF9900;" | 4 970 ||style="background-color: #FF9900;" | 11 748
|-
| '''5''' ||style="background-color: #bbffff;" | '''42''' ||style="background-color: #bbffff;" | '''170''' ||style="background-color: #FF9900;" | 336 ||style="background-color: #bbffff;" | '''2 730''' ||style="background-color: #FF9900;" | 3 110 ||style="background-color: #FF9900;" | 9 234 ||style="background-color: #FF9900;" | 27 936 ||style="background-color: #FF9900;" | 90 068
|-
| '''6''' ||style="background-color: #bbffff;" | '''62''' ||style="background-color: #bbffff;" | '''312''' ||style="background-color: #FF9900;" | 684 ||style="background-color: #bbffff;" | '''7 812''' ||style="background-color: #66ff66;" | 8 310 ||style="background-color: #FF9900;" | 29 790 ||style="background-color: #FF9900;" | 117 360 ||style="background-color: #FF9900;" | 452 032
|-
| '''7''' ||style="background-color: #ff0000;" | '''80''' ||style="background-color: #CC6600;" | 346 ||style="background-color: #FF9900;" | 1 134 ||style="background-color: #CC6600;" | 8 992 ||style="background-color: #66ff66;" | 23 436 ||style="background-color: #FF9900;" | 80 940 ||style="background-color: #FF9900;" | 400 160 ||style="background-color: #FF9900;" | 1 987 380
|-
| '''8''' ||style="background-color: #bbffff;" | '''114''' ||style="background-color: #bbffff;" | '''800''' ||style="background-color: #FF9900;" | 1 710 ||style="background-color: #bbffff;" | '''39 216''' ||style="background-color: #66ff66;" | 40 586 ||style="background-color: #FF9900;" | 201 480 ||style="background-color: #FF9900;" | 1 091 232 ||style="background-color: #FF9900;" | 6 927 210
|-
| '''9''' || style="background-color: #bbffff;" | '''146''' ||style="background-color: #bbffff;" | '''1 170''' ||style="background-color: #FF9900;" | 2 496 ||style="background-color: #bbffff;" | '''74 898''' ||style="background-color: #66ff66;" | 117 648 ||style="background-color: #FF9900;" | 449 480 ||style="background-color: #FF9900;" | 2 961 536 ||style="background-color: #FF9900;" | 20 017 260
|-
| '''10''' || style="background-color: #bbffff;" | '''182''' ||style="background-color: #bbffff;" | '''1 640''' ||style="background-color: #66ff66;" | 4 000 ||style="background-color: #bbffff;" | '''132 860''' ||style="background-color: #66ff66;" | 224 694 ||style="background-color: #66ff66;" | 1 176 480 ||style="background-color: #FF9900;" | 7 057 400 ||style="background-color: #FF9900;" | 50 331 156
|- 97 386 380
| '''11''' ||style="background-color: yellow;" | 190 || style="background-color: #CC6600;" | 1 734 ||style="background-color: #66ff66;" | 5 850 ||style="background-color: #CC6600;" | 142 464 ||style="background-color: #66ff66;" | 398 580 ||style="background-color: #66ff66;" | 2 246 940 || style="background-color: #FF9900;" | 15 200 448 ||style="background-color: #FF9900;" | 130 592 354
|-
| '''12''' ||style="background-color: #bbffff;" | '''266''' ||style="background-color: #bbffff;" | '''2 928''' ||style="background-color: #66ff66;" | 8 200||style="background-color: #bbffff;" | '''354 312''' ||style="background-color: #66ff66;" | 664 300 ||style="background-color: #66ff66;" | 4 650 100 ||style="background-color: #FF9900;" | 30 001 152 ||style="background-color: #FF9900;" | 300 383 050
|-
| '''13''' ||style="background-color: pink;" | 274 ||style="background-color: #CC6600;" | 3 064 ||style="background-color: #66ff66;" |11 480 ||style="background-color: #CC6600;" | 374 452 ||style="background-color: #66ff66;" | 1 062 936 ||style="background-color: #66ff66;" | 5 314 680 ||style="background-color: #FF9900;" | 50 990 610 ||style="background-color: #FF9900;" | 617 330 936
|-
| '''14''' ||style="background-color: #bbffff;" | '''366''' ||style="background-color: #bbffff;" | '''4 760''' ||style="background-color: #66ff66;" | 14 760 ||style="background-color: #bbffff;" | '''804 468''' ||style="background-color: #66ff66;" | 1 771 560 ||style="background-color: #66ff66;" | 14 172 480 ||style="background-color: #FF9900;" |95 087 738 ||style="background-color: #FF9900;" | 1 213 477 190
|-
| '''15''' || style="background-color: pink;" | 374 || style="background-color: #CC6600;" | 4 946 ||style="background-color: #66ff66;" | 20 496 || style="background-color: #CC6600;" | 842 048 ||style="background-color: #66ff66;" | 2 480 184 || style="background-color: #66ff66;" | 14 172 480 ||style="background-color: #FF9900;" | 168 016 334 ||style="background-color: #FF9900;" | 2 300 326 510
|-
| '''16''' ||style="background-color: #CC6600;" | 394 ||style="background-color: #CC6600;" | 5 134 ||style="background-color: #66ff66;" | 27 300 || style="background-color: #CC6600;" | 884 062 || style="background-color: #66ff66;" | 4 022 340 ||style="background-color: #66ff66;" | 36 201 060 ||style="background-color: #FF9900;" | 288 939 118 ||style="background-color: #FF9900;" | 4 119 507 330
|}
</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: #bbffff; text-align: center;" | * || Bipartite Moore graphs (optimal).
|-
|style="background-color: #CC6600; text-align: center;" | * || Graph duplications found by C. Delorme and G. Farhi.
|-
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by C. Delorme, J. Gómez, and J. J. Quisquater.
|-
|style="background-color: #0066CC; text-align: center;" | * || Optimal graph found by R. Bar-Yehuda and T. Etzion and by J. Bond and C. Delorme.
|-
|style="background-color: #993300; text-align: center;" | * || Graph found independently by M. Conder and R. Nedela., by C. Delorme, J. Gómez, and J. J. Quisquater and by Eyal Loz.
|-
|style="background-color: #ff0000; text-align: center;" | * || Graphs found independently by Paul Hafner and by Eyal Loz.
|-
|style="background-color: #FF9900; text-align: center;" | * || Graphs found by Eyal Loz as part of the joint project ''The degree/diameter problem for several classes of graphs'' by E. Loz, H. Pérez-Rosés and G. Pineda-Villavicencio.
|-
|style="background-color: yellow; text-align: center;" | * || Graphs found by R. Feria-Puron, M. Miller and G. Pineda-Villavicencio and independently by G. Araujo and N. López.
|-
|style="background-color: pink; text-align: center;" | * || Graphs found by G. Araujo and N. López.
|}
</center>

==Lowering and/or setting upper bounds for ''Nb(d,k)''==

The Moore bound can be reached in some cases, but not always in general. Some theoretical work was done to determine the lowest upper bounds. In this direction reserachers have been interested in bipartite graphs of maximum degree ''d'', diameter ''k'' and order ''Mb(d,k)-δ'' for small ''δ''. The parameter ''δ'' is called the defect. Such graphs are called bipartite ''(d,k,-δ)''-graphs.

The bipartite ''(d,k,-2;)''-graphs constitute the first interesting family of graphs to be studied. When ''d≥3'' and ''k=2'', bipartite ''(d,k,-2)''-graphs are the [http://en.wikipedia.org/wiki/Complete_bipartite_graph complete bipartite graphs] with partite sets of orders ''p'' and ''q'', where either ''p=q=d-1'' or ''p=d'' and ''q=d-2''. For ''d≥3'' and ''k≥3'' only two such graphs are known; a unique bipartite ''(3, 3,-2)''-graph and a unique bipartite ''(4, 3,-2)''-graph.

Studies on bipartite ''(d,k,-2;)''-graphs have been carried out by Charles Delorme, Leif Jorgensen, Mirka Miller and Guillermo Pineda-Villavicencio. They proved several necessary conditions for the existence of bipartite ''(d,3,-2;)''-graphs, the uniqueness of the two known bipartite ''(d,k,-2;)''-graphs for ''d≥3'' and ''k≥3'', and the non-existence of bipartite ''(d,k,-2;)''-graphs for ''d≥3'' and ''k≥4''.

===Lowest known upper bounds and the percentage of the order of the largest known bipartite graphs===

<center>
{| border="1"
| '''<math>d</math>\<math>k</math>'''|| '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''14'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''30'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|'''56'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''126'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|248
|-
|67.74%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|504
|-
|50.79%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|1016
|-
|49.80%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|2040
|-
|39.21%
|}
|-
|'''4'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''26'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''80'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|236
|-
|67.79%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''728'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|2180
|-
|38.53%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|6554
|-
|33.32%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|19676
|-
|25.25%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|59042
|-
|19.89%
|}
|-
|'''5'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''42'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''170'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|676
|-
|49.70%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''2730'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|10916
|-
|28.49%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|43684
|-
|21.13%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|174756
|-
|15.98%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|699044
|-
|12.88%
|}
|-
|'''6'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''62'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''312'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|1556
|-
|43.95%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''7812'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|39056
|-
|21.27%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|195306
|-
|15.25%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|976556
|-
|12.01%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|4882806
|-
|9.25%
|}
|-
|'''7'''
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|'''80'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|518
|-
|66.79%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|3106
|-
|36.50%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|18662
|-
|48.18%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|111968
|-
|20.92%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|671840
|-
|12.04%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|4031072
|-
|9.92%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|24186464
|-
|8.21%
|}
|-
|'''8'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''114'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''800'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|5596
|-
|30.55%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''39216'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|274508
|-
|14.78%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|1921594
|-
|10.48%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|13451196
|-
|8.11%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|94158410
|-
|7.35%
|}
|-
|'''9'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''146'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''1170'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|9356
|-
|26.67%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''74898'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|599180
|-
|19.63%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|4793484
|-
|9.37%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|38347916
|-
|7.72%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|306783372
|-
|6.52%
|}
|-
|'''10'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''182'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''1640'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|14756
|-
|27.10%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''132860'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|1195736
|-
|18.79%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|10761674
|-
|10.93%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|96855116
|-
|7.28%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|871696094
|-
|5.77%
|}
|-
|'''11'''
| align="center" |
{| border="2" style="background:rgb(100,100,255);"
|220
|-
|86.36%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|2222
|-
|78.03%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|22216
|-
|26.33%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|222222
|-
|64.10%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|2222216
|-
|17.93%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|22222216
|-
|10.11%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|222222216
|-
|6.84%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|2222222216
|-
|5.87%
|}
|-
|'''12'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''266'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''2928'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|32204
|-
|25.46%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''354312'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|3897428
|-
|17.04%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|42871770
|-
|10.84%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|471589532
|-
|6.36%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|5187484914
|-
|5.79%
|}
|-
|'''13'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|314
|-
|85.98%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|3770
|-
|81.27%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|45236
|-
|25.37%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|542906
|-
|68.97%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|6514868
|-
|16.31%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|78178484
|-
|6.79%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|938141876
|-
|5.43%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|11257702580
|-
|5.48%
|}
|-
|'''14'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''366'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''4760'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|61876
|-
|23.85%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|'''804468'''
|-
|100%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|10458080
|-
|16.93%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|135955114
|-
|10.42%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|1767416556
|-
|5.38%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|22976415302
|-
|5.28%
|}
|-
|'''15'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|422
|-
|87.67%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|5910
|-
|83.68%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|82736
|-
|24.77%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|1158390
|-
|72.69%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|16217456
|-
|15.29%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|227044464
|-
|0%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|3178622576
|-
|5.28%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|44500716144
|-
|5.17%
|}
|-
|'''16'''
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|482
|-
|81.74%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|7232
|-
|70.99%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|108476
|-
|25.16%
|}
| align="center" |
{| border="2" style="background:rgb(40,40,255);"
|1627232
|-
|54.32%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|24408476
|-
|16.47%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|366127226
|-
|9.88%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|5491908476
|-
|5.26%
|}
| align="center" |
{| border="2" style="background:rgb(210,210,255);"
|82378627226
|-
|5%
|}
|}
</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: rgb(40,40,255); text-align: center;" | * || Moore bound.
|-
|style="background-color: rgb(100,100,255); text-align: center;" | * || Moore bound minus 2.
|-
|style="background-color: rgb(210,210,255); text-align: center;" | * || Moore bound minus 6.
|-
|}
</center>

==References==
* Conder, M.; Nedela, R. (2006), "A more detailed classification of symmetric cubic graphs", preprint.
* Bar-Yehuda, R.; Etzion, T. (1992), "Connections between two cycles - a new design of dense processor interconnection networks", Discrete Applied Mathematics 37-38.
* Bond, J.; Delorme, C. (1988), "New large bipartite graphs with given degree and diameter", Ars Combinatoria 25C: 123-132.
* Bond, J.; Delorme, C. (1993), "A note on partial Cayley graphs", Discrete Mathematics 114 (1-3): 63--74, doi:10.1016/0012-365X(93)90356-X.
* Delorme, C. (1985), "Grands graphes de degré et diamètre donnés", European Journal of Combinatorics 6: 291-302.
* Delorme, C. (1985), "Large bipartite graphs with given degree and diameter", Journal of Graph Theory 8: 325-334.
* Delorme, C.; Farhi, G. (1984), "Large graphs with given degree and diameter Part I", IEEE Transactions on Computers C-33: 857-860.
* Delorme, C.; Gómez (2002), "Some new large compound graphs", European Journal of Combinatorics 23 (5): 539-547, doi:10.1006/eujc.2002.0581.
* Delorme, C.; Gómez, J.; Quisquater, J. J., "On large bipartite graphs", submitted.
* Delorme, C.; Jorgensen, L.; Miller, M.; Pineda-Villavicencio, G., "On bipartite graphs of diameter 3 and defect 2", Journal of Graph Theory 61 (2009), no. 4, 271-288.
* Delorme, C.; Jorgensen, L.; Miller, M.; Pineda-Villavicencio, G., "On bipartite graphs of defect 2", European Journal of Combinatorics 30 (2009), no. 4, 798-808.
*Pineda-Villavicencio, G., Non-existence of bipartite graphs of diameter at least 4 and defect 2, Journal of Algebraic Combinatorics 34 (2011), no. 2, 163-182.
* Miller, M.; Širáň, J. (2005), "Moore graphs and beyond: A survey of the degree/diameter problem", Electronic Journal of Combinatorics Dynamic survey D, [http://www.combinatorics.org/Surveys/ds14.pdf PDF version].

==External links==
* [http://www.eyal.com.au/wiki/The_Degree/Diameter_Problem Eyal Loz's] Degree-Diameter problem page, including adjacency lists for bipartite graphs smaller than 20,000 found as a part of the project ''The degree/diameter problem for several classes of graphs''.

[[Category:The Degree/Diameter Problem]]

Tables and Results

2019-09-16T12:14:59Z

Grahame: /* Table 2: bounds for trivalent cages */

==Trivalent cages==
===Table 1: known trivalent cages===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''g'' ''' || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''|| style="background-color: #cccccc;" | '''8'''|| style="background-color: #cccccc;" | '''9'''|| style="background-color: #cccccc;" | '''10'''|| style="background-color: #cccccc;" | '''11'''|| style="background-color: #cccccc;" | '''12'''
|-
|style="background-color: #cccccc;"| ''' ''n(3,g)'' ''' || 10 ||14 ||24 ||30 ||58 ||70 ||112 ||126
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 1|| 1|| 18|| 3|| 1|| 1
|}
</center>

===Table 2: bounds for trivalent cages===
Optimal graphs are marked in bold
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Girth ''g'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' ||style="background-color: #cccccc;"| ''' Number of cages''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"| 5 ||style="background-color: #cccccc;"| 10 || '''10''' ||1 ||[http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"| 6 ||style="background-color: #cccccc;"| 14 ||'''14'''|| 1 ||[http://en.wikipedia.org/wiki/Heawood_graph Heawood]
|-
|style="background-color: #cccccc;"| 7 ||style="background-color: #cccccc;"| 24 || '''24''' ||1 ||McGee
|-
|style="background-color: #cccccc;"| 8 ||style="background-color: #cccccc;"| 30 ||'''30'''|| 1 ||[http://en.wikipedia.org/wiki/Tutte_eight_cage Tutte]
|-
|style="background-color: #cccccc;"| 9 ||style="background-color: #cccccc;"| 58 ||'''58''' ||18 ||Brinkmann-McKay-Saager
|-
|style="background-color: #cccccc;"| 10 ||style="background-color: #cccccc;"| 70 ||'''70''' ||3 ||O’Keefe-Wong
|-
|style="background-color: #cccccc;"| 11 ||style="background-color: #cccccc;"| 112 || '''112'''|| 1 ||McKay-Myrvold; Balaban
|-
|style="background-color: #cccccc;"| 12 ||style="background-color: #cccccc;"| 126 || '''126'''|| 1|| Benson
|-
|style="background-color: #cccccc;"| 13 ||style="background-color: #cccccc;"| 202 ||272|| ||McKay-Myrvold; Hoare
|-
|style="background-color: #cccccc;"| 14 ||style="background-color: #cccccc;"| 258 ||384 ||||McKay; Exoo
|-
|style="background-color: #cccccc;"| 15 ||style="background-color: #cccccc;"| 384 ||620|||| Biggs
|-
|style="background-color: #cccccc;"| 16 ||style="background-color: #cccccc;"| 512 ||960|| ||Exoo
|-
|style="background-color: #cccccc;"| 17 ||style="background-color: #cccccc;"| 768 ||2176|| ||Exoo
|-
|style="background-color: #cccccc;"| 18 ||style="background-color: #cccccc;"| 1024 ||2560|| ||Exoo
|-
|style="background-color: #cccccc;"| 19 ||style="background-color: #cccccc;"| 1536 ||4324|||| Hoare, H(47)
|-
|style="background-color: #cccccc;"| 20 ||style="background-color: #cccccc;"| 2048 || 5376 || ||Exoo
|-
|style="background-color: #cccccc;"| 21 ||style="background-color: #cccccc;"| 3072 ||16028 || ||Exoo
|-
|style="background-color: #cccccc;"| 22 ||style="background-color: #cccccc;"| 4096 || 16206|| || Biggs-Hoare, S(73)
|-
|style="background-color: #cccccc;"| 23 ||style="background-color: #cccccc;"| 6144 ||49326 || ||Exoo
|-
|style="background-color: #cccccc;"| 24 ||style="background-color: #cccccc;"| 8192 || 49608 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 25 ||style="background-color: #cccccc;"| 12288 ||108906|| || Exoo
|-
|style="background-color: #cccccc;"| 26 ||style="background-color: #cccccc;"| 16384 || 109200 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 27 ||style="background-color: #cccccc;"| 24576 ||285852 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 28 ||style="background-color: #cccccc;"| 32768 ||415104|| || Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 29 ||style="background-color: #cccccc;"| 49152 ||1141484|| || Exoo-Jajcay
|-
|style="background-color: #cccccc;"| 30 ||style="background-color: #cccccc;"| 65536 ||1143408|| || Exoo-Jajcay
|-
|style="background-color: #cccccc;"| 31 ||style="background-color: #cccccc;"| 98304 ||3649794 || ||Bray-Parker-Rowley
|-
|style="background-color: #cccccc;"| 32 ||style="background-color: #cccccc;"| 131072 ||3650304|| || Bray-Parker-Rowley
|}
</center>

==Cages of girth 5 and 6==
Optimal graphs are marked in bold.
===Table 1: known cages of Girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' ''k'' ''' || style="background-color: #cccccc;" | 3 || style="background-color: #cccccc;" | 4 || style="background-color: #cccccc;" | '''5''' || style="background-color: #cccccc;" | '''6'''|| style="background-color: #cccccc;" | '''7'''
|-
|style="background-color: #cccccc;"| ''' ''n(k,5)'' ''' || 10 || 19|| 30|| 40|| 50
|-
|style="background-color: #cccccc;"| '''number of cages'''|| 1|| 1|| 4|| 1|| 1
|}
</center>

===Table 2: bounds for cages of girth 5===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"|10 ||'''10'''|| [http://en.wikipedia.org/wiki/Petersen_graph Petersen]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|19 ||'''19'''|| Robertson
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|30 ||'''30'''|| Robertson-Wegner-Wong
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|40 ||'''40'''|| Wong
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|50||''' 50'''|| [http://en.wikipedia.org/wiki/Hoffman-Singleton_graph Hoﬀman-Singleton]
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|67|| 80|| Royle
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|86|| 96 ||Jørgensen
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|103|| 126 ||Exoo
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|124 ||156|| Jørgensen
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|147|| 203|| Exoo
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|174 ||240 ||Exoo
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|199 ||288 ||Jørgensen
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|230 ||312 ||Jørgensen
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|259 ||336|| Jørgensen
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|294|| 448 ||Schwenk
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|327|| 480|| Schwenk
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|364|| 512|| Schwenk
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|403|| 576 ||Jørgensen
|}
</center>

===Table 3: bounds for cages of girth 6===
<center>
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"
|-
|style="background-color: #cccccc;"| ''' Degree ''k'' ''' ||style="background-color: #cccccc;"| ''' Lower bound ''' ||style="background-color: #cccccc;"| ''' Upper bound ''' || style="background-color: #cccccc;"| ''' Author or description '''
|-
|style="background-color: #cccccc;"|3 ||style="background-color: #cccccc;"| 14 ||'''14'''|| [http://en.wikipedia.org/wiki/Projective_plane Projective Plane]
|-
|style="background-color: #cccccc;"|4 ||style="background-color: #cccccc;"|26 ||'''26'''|| Projective Plane
|-
|style="background-color: #cccccc;"|5 ||style="background-color: #cccccc;"|42 ||'''42'''|| Projective Plane
|-
|style="background-color: #cccccc;"|6 ||style="background-color: #cccccc;"|62 ||'''62'''|| Projective Plane
|-
|style="background-color: #cccccc;"|7 ||style="background-color: #cccccc;"|90 ||'''90'''|| O’Keefe-Wong
|-
|style="background-color: #cccccc;"|8 ||style="background-color: #cccccc;"|114 ||'''114'''|| Projective Plane
|-
|style="background-color: #cccccc;"|9 ||style="background-color: #cccccc;"|146 ||'''146'''|| Projective Plane
|-
|style="background-color: #cccccc;"|10 ||style="background-color: #cccccc;"|182 ||'''182'''|| Projective Plane
|-
|style="background-color: #cccccc;"|11 ||style="background-color: #cccccc;"|224 ||240|| Wong
|-
|style="background-color: #cccccc;"|12 ||style="background-color: #cccccc;"|266|| 266|| Projective Plane
|-
|style="background-color: #cccccc;"|13 ||style="background-color: #cccccc;"|314 ||336|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|14 ||style="background-color: #cccccc;"|366 ||'''366'''|| Projective Plane
|-
|style="background-color: #cccccc;"|15 ||style="background-color: #cccccc;"|422 ||462|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|16 ||style="background-color: #cccccc;"|482 ||504|| Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|17 ||style="background-color: #cccccc;"|546 ||'''546'''|| Projective Plane
|-
|style="background-color: #cccccc;"|18 ||style="background-color: #cccccc;"|614 ||'''614'''|| Projective Plane
|-
|style="background-color: #cccccc;"|19 ||style="background-color: #cccccc;"|686 ||720 ||Abreu-Funk-Labbate-Napolitano
|-
|style="background-color: #cccccc;"|20 ||style="background-color: #cccccc;"|762 ||'''762'''|| Projective Plane
|}
</center>

[[Category:The Cage Problem]]

The Degree Diameter Problem for Circulant Graphs

2019-07-29T08:29:13Z

Grahame: /* Table of the orders of the largest known circulant 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: 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;" | 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

2019-03-28T09:15:12Z

Grahame: /* 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
|}
</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%
|}

|}
</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

2019-03-28T09:13:15Z

Grahame: /* Table of the orders of the largest known circulant 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
|}
</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
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|42%
|}
| 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%
|}

|}
</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

2019-03-13T10:11:06Z

Grahame: /* 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;" | 544 ||style="background-color: gold;" | 1 886 ||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
|}
</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:rgb(180,180,255);"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|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
|-
|56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|42%
|}
| 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%
|}

|}
</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

2019-03-13T10:08:28Z

Grahame: /* Table of the orders of the largest known circulant 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;" | 544 ||style="background-color: gold;" | 1 886 ||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
|}
</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:rgb(180,180,255);"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|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
|-
|0%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|0%
|}
| 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%
|}

|}
</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

2019-03-12T17:30:21Z

Grahame: /* Table of the orders of the largest known circulant 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;" | 544 ||style="background-color: gold;" | 1 874 ||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
|}
</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:rgb(180,180,255);"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|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
|-
|0%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|0%
|}
| 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%
|}

|}
</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

2019-03-12T14:12:08Z

Grahame: /* Table of the orders of the largest known circulant 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;" | 544 ||style="background-color: gold;" | 1 870 ||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
|}
</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:rgb(180,180,255);"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|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
|-
|0%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|0%
|}
| 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%
|}

|}
</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

2019-03-11T16:08:49Z

Grahame: /* Table of the orders of the largest known circulant 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;" | 544 || | ||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
|}
</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:rgb(180,180,255);"
|3 608
|-
|88%
|}
| align="center" |
{| border="2" style="background:rgb(180,180,255);"
|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
|-
|0%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
| 4 482
|-
|0%
|}
| 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%
|}

|}
</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 General Graphs

2019-02-11T19:09:48Z

Grahame: /* Table of the orders of the largest known graphs for the undirected degree diameter problem */

==Introduction==
The '''degree/diameter problem for general graphs''' can be stated as follows:

''Given natural numbers ''d'' and ''k'', find the largest possible number ''N(d,k)'' of vertices in a graph of maximum degree ''d'' and diameter ''k''.''

In attempting to settle the values of ''N(d,k)'', research activities in this problem have follow the following two directions:

*Increasing the lower bounds for ''N(d,k)'' by constructing ever larger graphs.

* Lowering and/or setting upper bounds for ''N(d,k)'' by proving the non-existence of graphs
whose order is close to the Moore bounds ''M(d,k)=(d(d-1)k-2)(d-2)-1''.

==Increasing the lower bounds for ''N(d,k)''==

In the quest for the largest known graphs many innovative approaches have been suggested. In a wide spectrum, we can classify these approaches into general (those producing graphs for many combinations of the degree and the diameter) and ad hoc (those devised specifically for producing graphs for few combinations of the degree and the diameter). Among the former, we have the constructions of [http://en.wikipedia.org/wiki/De_Bruijn_graph De Bruijn graphs] and [http://en.wikipedia.org/wiki/Kautz_graph Kautz graphs], while among the latter, we have the star product, the voltage assigment technique and graph compunding. For information on the state-of -the-art of this research stream, the interested reader is referred to the survey by Miller and Širáň.

Below is the table of the largest known graphs (as of September 2009) in the undirected [[The Degree/Diameter Problem | degree diameter problem]] for graphs of [http://en.wikipedia.org/wiki/Degree_(graph_theory) degree] at most 3 ≤ ''d'' ≤ 20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2 ≤ ''k'' ≤ 10. Only a few of the graphs in this table are known to be optimal (marked in bold), and thus, finding a larger graph that is closer in order (in terms of the size of the vertex set) to the [http://en.wikipedia.org/wiki/Moore_graph Moore bound] is considered an [http://en.wikipedia.org/wiki/Open_problem open problem]. Some general constructions are known for values of ''d'' and ''k'' outside the range shown in the table.

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

<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: red;" | '''10''' ||style="background-color: blue;" | '''20''' ||style="background-color: blue;" | '''38''' ||style="background-color: #66cc66;" | 70 ||style="background-color: #ffff00;" | 132 ||style="background-color: #ffff00;" | 196 ||style="background-color: #ddff00;" | 360 ||style="background-color: #ffff00;" | 600 ||style="background-color: #ffcc99;" | 1 250
|-

| '''4''' ||style="background-color: blue;" | '''15''' ||style="background-color: #999900;" | 41 ||style="background-color: #ffff00;" | 98 ||style="background-color: #81BEF7; text-align: center;" | 364 ||style="background-color: #666666;" | 740 ||style="background-color: #FF9900;" | 1 320 ||style="background-color: #FF9900;" | 3 243 ||style="background-color: #FF9900;" | 7 575 ||style="background-color: #FF9900;" | 17 703
|-
| '''5''' ||style="background-color: blue;" | '''24''' ||style="background-color: #ffff00;" | 72 ||style="background-color: #ffff00;" | 212 ||style="background-color: #FF9900;" | 624 ||style="background-color: #00ff7f;" | 2 772 ||style="background-color: #FF9900;" | 5 516 ||style="background-color: #FF9900;" | 17 030 ||style="background-color: #FF9900;" | 57 840 ||style="background-color: #FF9900;" | 187 056
|-
| '''6''' ||style="background-color: #336666;" | '''32''' ||style="background-color: #ffff00;" | 111 ||style="background-color: #FF9900;" | 390 ||style="background-color: #FF9900;" | 1 404 ||style="background-color: #00ff7f;" | 7 917 ||style="background-color: #FF9900;" | 19 383 ||style="background-color: #FF9900;" | 76 461 ||style="background-color: #aa8268;" | 331 387 ||style="background-color: #FF9900;" | 1 253 615
|-
| '''7''' ||style="background-color: red;" | '''50''' ||style="background-color: #ffff00;" | 168 ||style="background-color: #bbffff;" | 672 ||style="background-color: #FF0066;" | 2 756 ||style="background-color: #FF9900;" | 11 988 ||style="background-color: #FF9900;" | 52 768 ||style="background-color: #FF9900;" | 249 660 ||style="background-color: #FF9900;" | 1 223 050 ||style="background-color: #FF9900;" | 6 007 230
|-
| '''8''' || style="background-color: #81BEF7; text-align: center;" |57 ||style="background-color: #993300;" | 253 ||style="background-color: #FF9900;" | 1 100 ||style="background-color: #FF9900;" | 5 060 ||style="background-color: #666633;" | 39 672 ||style="background-color: #FF9900;" | 131 137 ||style="background-color: #FF9900;" | 734 820 ||style="background-color: #FF9900;" | 4 243 100 ||style="background-color: #FF9900;" | 24 897 161
|-
| '''9''' ||style="background-color: #6666ff; text-align: center;" | 74 ||style="background-color: #81BEF7; text-align: center;" | 585 ||style="background-color: #FF9900;" | 1 550 ||style="background-color: #aa8268;" | 8 268 ||style="background-color: #00ff7f;" | 75 893 ||style="background-color: #FF9900;" | 279 616 ||style="background-color: #aa8268;" | 1 697 688||style="background-color: #FF9900;" | 12 123 288 ||style="background-color: #FF9900;" | 65 866 350
|-
| '''10''' ||style="background-color: #81BEF7; text-align: center;" | 91 || style="background-color: #6666ff; text-align: center;" |650 ||style="background-color: #FF9900;" | 2 286 ||style="background-color: #FF9900;" | 13 140 ||style="background-color: #666633;" | 134 690 ||style="background-color: #FF9900;" | 583 083 ||style="background-color: #FF9900;" | 4 293 452 ||style="background-color: #FF9900;" | 27 997 191 ||style="background-color: #FF9900;" | 201 038 922
|-
| '''11''' ||style="background-color: #ffff00;" | 104 || style="background-color: #6666ff; text-align: center;" |715 ||style="background-color: #cccccc; text-align: center;" | 3 200 ||style="background-color: #FF9900;" | 19 500 ||style="background-color: #cccccc; text-align: center;" | 156 864 ||style="background-color: #FF9900;" | 1 001 268 ||style="background-color: #FF9900;" | 7 442 328 || style="background-color: #FF9900;" | 72 933 102 ||style="background-color: #FF9900;" | 600 380 000
|-
| '''12''' ||style="background-color: #81BEF7; text-align: center;" | 133 ||style="background-color: #666633;" | 786 ||style="background-color: #3399cc; text-align: center;" | 4 680 ||style="background-color: #FF9900;" | 29 470||style="background-color: #00ff7f;" | 359 772 ||style="background-color: #FF9900;" | 1 999 500 ||style="background-color: #FF9900;" | 15 924 326 ||style="background-color: #FF9900;" | 158 158 875 ||style="background-color: #FF9900;" | 1 506 252 500
|-
| '''13''' ||style="background-color: #ff99ff;" | 162 ||style="background-color: #666633;" | 851 ||style="background-color: #cccccc; text-align: center;" | 6 560 ||style="background-color: #FF9900;" |40 260 || style="background-color: #cccccc; text-align: center;" | 531 440 ||style="background-color: #FF9900;" | 3 322 080 ||style="background-color: #FF9900;" | 29 927 790 ||style="background-color: #FF9900;" | 249 155 760 ||style="background-color: #FF9900;" | 3 077 200 700
|-
| '''14''' ||style="background-color: #81BEF7; text-align: center;" | 183 ||style="background-color: #666633;" | 916 ||style="background-color: #cccccc; text-align: center;" | 8 200 ||style="background-color: #FF9900;" | 57 837 ||style="background-color: #00ff7f;" | 816 294 ||style="background-color: #999999; text-align: center;" | 6 200 460 ||style="background-color: #FF9900;" | 55 913 932 ||style="background-color: #FF9900;" | 600 123 780 ||style="background-color: #FF9900;" | 7 041 746 081
|-
| '''15''' ||style="background-color: #187eac; text-align: center;" | 187 ||style="background-color: #81BEF7; text-align: center;" | 1 215 ||style="background-color: #cccccc; text-align: center;" | 11 712 ||style="background-color: #FF9900;" | 76 518 || style="background-color: #cccccc; text-align: center;" |1 417 248 ||style="background-color: #FF9900;" | 8 599 986 ||style="background-color: #FF9900;" | 90 001 236 ||style="background-color: #FF9900;" | 1 171 998 164 ||style="background-color: #FF9900;" | 10 012 349 898
|-
| '''16''' ||style="background-color: #99FF00;" | 200 ||style="background-color: #81BEF7; text-align: center;" | 1 600 ||style="background-color: #cccccc; text-align: center;" | 14 640 ||style="background-color: #81BEF7; text-align: center;" | 132 496 ||style="background-color: #cccccc; text-align: center;" | 1 771 560 || style="background-color: #999999; text-align: center;" |14 882 658 ||style="background-color: #FF9900;" | 140 559 416 ||style="background-color: #FF9900;" | 2 025 125 476 ||style="background-color: #FF9900;" | 12 951 451 931

|-
| '''17''' ||style="background-color: #cc0033;" | 274 ||style="background-color: #ff6600;" | 1 610 ||style="background-color: #ff6600;" | 19 040 ||style="background-color: #ff6600;" | 133 144 || style="background-color: #ff6600;" | 3 217 872 || style="background-color: #ff6600;" | 18 495 162 ||style="background-color: #ff6600;" | 220 990 700 ||style="background-color: #ff6600;" | 3 372 648 954 ||style="background-color: #ff6600;" | 15 317 070 720

|-
| '''18''' ||style="background-color: #cc0033;" | 307 ||style="background-color: #ff6600;" | 1 620 ||style="background-color: #ff6600;" | 23 800 ||style="background-color: #ff6600;" | 171 828 || style="background-color: #ff6600;" | 4 022 340 || style="background-color: #ff6600;" | 26 515 120 ||style="background-color: #ff6600;" | 323 037 476 ||style="background-color: #ff6600;" | 5 768 971 167 ||style="background-color: #ff6600;" | 16 659 077 632

|-
| '''19''' ||style="background-color: #ff99ff;" | 338 ||style="background-color: #ff6600;" | 1 638 ||style="background-color: #ff6600;" | 23 970 ||style="background-color: #ff6600;" | 221 676 || style="background-color: #ff6600;" | 4 024 707 || style="background-color: #ff6600;" | 39 123 116 ||style="background-color: #ff6600;" | 501 001 000 ||style="background-color: #ff6600;" | 8 855 580 344 ||style="background-color: #ff6600;" | 18 155 097 232

|-
| '''20''' ||style="background-color: #cc0033;" | 381 ||style="background-color: #ff6600;" | 1 958 ||style="background-color: #ff6600;" | 34 952 ||style="background-color: #ff6600;" | 281 820 || style="background-color: #ff6600;" | 8 947 848 || style="background-color: #ff6600;" | 55 625 185 ||style="background-color: #ff6600;" | 762 374 779 ||style="background-color: #ff6600;" | 12 951 451 931 ||style="background-color: #ff6600;" | 78 186 295 824

|}
</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: red; text-align: center;" | * || The [http://en.wikipedia.org/wiki/Petersen_graph Petersen] and [http://en.wikipedia.org/wiki/Hoffman–Singleton_graph Hoffman–Singleton] graphs.
|-
|style="background-color: blue; text-align: center;" | * || Other non Moore but optimal graphs.
|-
|style="background-color: #999900; text-align: center;" | * || Graph found by J. Allwright.
|-
|style="background-color: #336666; text-align: center;" | * || Graph found by G. Wegner.
|-
|style="background-color: #ffff00; text-align: center;" | * || Graphs found by G. Exoo.
|-
|style="background-color: #ff99ff; text-align: center;" | * || Family of graphs found by B. D. McKay, M. Miller and J. Širáň. More details are available in a paper by the authors.
|-
|style="background-color: #666633; text-align: center;" | * || Graphs found by J. Gómez.
|-
|style="background-color: #993300; text-align: center;" | * || Graph found by M. Mitjana and F. Comellas. This graph was also found independently by M. Sampels.
|-
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by C. Delorme.
|-
|style="background-color: #6666ff; text-align: center;" | * || Graphs found by C. Delorme and G. Farhi.
|-
|style="background-color: #187eac; text-align: center;" | * || Graphs found by E. Canale. (2012)
|-
|style="background-color: #3399cc; text-align: center;" | * || Graph found by J. C. Bermond, C. Delorme, and G. Farhi
|-
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by J. Gómez and M. A. Fiol.
|-
|style="background-color: #999999; text-align: center;" | * || Graphs found by J. Gómez, M. A. Fiol, and O. Serra.
|-
|style="background-color: #66cc66; text-align: center;" | * || Graph found by M.A. Fiol and J.L.A. Yebra.
|-
|style="background-color: #666666; text-align: center;" | * || Graph found by F. Comellas and J. Gómez.
|-
|style="background-color: #ddff00; text-align: center;" | * || Graph found by Jianxiang Chen.
|-
|style="background-color: #00ff7f; text-align: center;" | * || Graphs found by G. Pineda-Villavicencio, J. Gómez, M. Miller and H. Pérez-Rosés. More details are available in a paper by the authors.
|-
|style="background-color: #FF9900; text-align: center;" | * || Graphs found by E. Loz. More details are available in a paper by E. Loz and J. Širáň.
|-
|style="background-color: #ff6600; text-align: center;" | * || Graphs found by E. Loz and G. Pineda-Villavicencio. More details are available in a paper by the authors.
|-
|style="background-color: #aa8268; text-align: center;" | * || Graphs found by A. Rodriguez. (2012)
|-
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by M. Sampels.
|-
|style="background-color: #FF0066; text-align: center;" | * || Graphs found by M. J. Dinneen and P. Hafner. More details are available in a paper by the authors.
|-
|style="background-color: #ffcc99; text-align: center;" | * || Graph found by M. Conder.
|-
|style="background-color: #cc0033; text-align: center;" | * || Graphs found by Brown, W. G. (1966).
|-
|style="background-color: #99FF00; text-align: center;" | * || Graph found by M. Abas. (2017). More details are available in a paper by the author.
|}
</center>

==Lowering and/or setting upper bounds for ''N(d,k)''==

As the Moore bound cannot be reached in general, some theoretical work has been done to determine the lowest possible upper bounds. In this direction reserachers have been interested in graphs of maximum degree ''d'', diameter ''k'' and order ''M(d,k)-δ'' for small ''δ''. The parameter ''δ'' is called the defect. Such graphs are called ''(d,k,-δ)''-graphs.

For ''δ=1'' the only ''(d,k,-1)''-graphs are the cycles on ''2k'' vertices. Erdös, Fajtlowitcz and Hoffman, who proved the non-existence of ''(d,2,-1)''-graphs for ''d≠3''. Then, Bannai and Ito, and also
independently, Kurosawa and Tsujii, proved the non-existence of ''(d,k,-1)''-graphs for ''d≥3'' and ''k≥3''.

For ''δ=2'', the ''(2,k,-2)''-graphs are the cycles on ''2k-1''. Considering ''d≥3'', only five graphs are known at present. Elspas found the unique ''(4,2,-2)''-graph and the unique ''(5,2,-2)''-graph, and credited Green with producing the unique ''(3,3,-2)''-graph. The other graphs are two non-isomorphic ''(3,2,-2)''-graphs.

When ''δ=2'', ''d≥3'' and ''k≥3'', not much is known about the existence or otherwise of ''(d,k,-2)''-graphs. In this context some known outcomes include the non-existence of ''(3,k,-2)''-graphs with ''k≥4'' by Leif Jorgensen, the non-existence of ''(4,k,-2)''-graphs with ''k≥3'' by Mirka Miller and Rino Simanjuntak, some structural properties of ''(5,k,-2)''-graphs with ''k≥3'' by Guillermo Pineda-Villavicencio and Mirka Miller, the obtaining of several necessary conditions for the existence of ''(d,2,-2)''-graphs with ''d≥3'' by Mirka Miller, Minh Nguyen and Guillermo Pineda-Villavicencio, and the non-existence of ''(d,2,-2)''-graphs for ''5<d<50'' by Jose Conde and Joan Gimbert.

For the case of ''δ≥3'' only a few works are known at present: the non-existence of ''(3,4,-4)''-graphs by Leif Jorgensen; the complete catalogue of ''(3,k,-4)''-graphs with ''k≥2'' by Guillermo Pineda-Villavicencio and Mirka Miller by proving the non-existence of ''(3,k,-4)''-graphs with ''k≥5'', the settlement of ''N(3,4)=M(3,4)=38'' by Buset; and the obtaining of ''N(6,2)=M(6,2)-5=32'' by Molodtsov. For more information, check the corresponding papers, and the survey by Miller and Širáň.

===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'''
|-
|'''3'''
| align="center" |
{| border="2" style="background:red;"
|10
|-
|100%
|}
| align="center" |
{| border="2" style="background:#ffff00;"
|20
|-
|100%
|}
| align="center" |
{| border="2" style="background:#336666;"
|38
|-
|100%
|}
| align="center" |
{| border="2" style="background:#999900;"
|92
|-
|76.08%
|}
| align="center" |
{| border="2" style="background:#999900;"
|188
|-
|70.21%
|}
| align="center" |
{| border="2" style="background:#999900;"
|380
|-
|51.57%
|}
| align="center" |
{| border="2" style="background:#999900;"
|764
|-
|43.97%
|}
| align="center" |
{| border="2" style="background:#999900;"
|1532
|-
|39.16%
|}
| align="center" |
{| border="2" style="background:#999900;"
|3068
|-
|40.74%
|}
|-
|'''4'''
| align="center" |
{| border="2" style="background:#ffff00;"
|15
|-
|100%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52
|-
|78.84%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|160
|-
|61.25%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|484
|-
|75.20%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1456
|-
|50.82%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4372
|-
|30.19%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13120
|-
|24.71%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|39364
|-
|19.24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|118096
|-
|14.99%
|}
|-
|'''5'''
| align="center" |
{| border="2" style="background:#ffff00;"
|24
|-
|100%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|104
|-
|69.23%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|424
|-
|50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1704
|-
|36.61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|6824
|-
|40.62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|27304
|-
|20.20%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|109224
|-
|15.59%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|436904
|-
|13.23%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1747624
|-
|10.70%
|}
|-
|'''6'''
| align="center" |
{| border="2" style="background:#336666;"
|32
|-
|100%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|186
|-
|59.67%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|936
|-
|41.66%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4686
|-
|29.96%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|23436
|-
|33.78%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|117186
|-
|16.54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|585936
|-
|13.04%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2929686
|-
|10.50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14648436
|-
|8.55%
|}
|-
|'''7'''
| align="center" |
{| border="2" style="background:red;"
|50
|-
|100%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|301
|-
|55.81%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1813
|-
|37.06%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|10885
|-
|25.31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|65317
|-
|18.35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|391909
|-
|13.46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2351461
|-
|10.61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|14108773
|-
|8.66%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|84652645
|-
|7.09%
|}
|-
|'''8'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|63
|-
|90.47%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|456
|-
|55.48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3200
|-
|34.37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|22408
|-
|22.58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|156864
|-
|25.29%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1098056
|-
|11.94%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7686400
|-
|9.56%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|53804808
|-
|7.88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|376633664
|-
|6.61%
|}
|-
|'''9'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|80
|-
|92.50%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|657
|-
|89.04%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5265
|-
|29.43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|42129
|-
|19.46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|337041
|-
|22.51%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2696337
|-
|10.37%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|21570705
|-
|7.87%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|172565649
|-
|7.02%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1380525201
|-
|4.77%
|}
|-
|'''10'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|99
|-
|91.91%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|910
|-
|71.42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8200
|-
|27.87%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|73810
|-
|17.80%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|664300
|-
|20.27%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5978710
|-
|9.75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|53808400
|-
|7.97%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|484275610
|-
|5.78%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4358480500
|-
|4.61%
|}
|-
|'''11'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|120
|-
|86.66%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1221
|-
|58.55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12221
|-
|26.18%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|122221
|-
|15.95%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1222221
|-
|12.83%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12222221
|-
|8.19%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|122222221
|-
|6.08%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1222222221
|-
|5.96%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12222222221
|-
|4.91%
|}
|-
|'''12'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|143
|-
|93%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1596
|-
|49.24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|17568
|-
|26.63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|193260
|-
|15.24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2125872
|-
|16.92%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|23384604
|-
|8.55%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|257230656
|-
|6.19%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2829537228
|-
|5.58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|31124909520
|-
|4.83%
|}
|-
|'''13'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|168
|-
|96.42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2041
|-
|41.69%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|24505
|-
|26.77%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|294073
|-
|13.69%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3528889
|-
|15.05%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|42346681
|-
|7.84%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|508160185
|-
|5.88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|6097922233
|-
|4.08%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|73175066809
|-
|4.20%
|}
|-
|'''14'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|195
|-
|93.84%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2562
|-
|35.75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|33320
|-
|24.60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|433174
|-
|13.35%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5631276
|-
|14.49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|73206602
|-
|8.46%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|951685840
|-
|5.87%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12371915934
|-
|4.85%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|160834907156
|-
|4.37%
|}
|-
|'''15'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|224
|-
|83.03%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3165
|-
|38.38%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|44325
|-
|26.42%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|620565
|-
|12.33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|8687925
|-
|16.31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|121630965
|-
|7.07%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1702833525
|-
|5.28%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|23839669365
|-
|4.91%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|333755371125
|-
|2.99%
|}
|-
|'''16'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|255
|-
|77.64%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3856
|-
|41.49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|57856
|-
|25.30%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|867856
|-
|15.26%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|13017856
|-
|13.60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|195267856
|-
|7.62%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2929017856
|-
|4.79%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|43935267856
|-
|4.60%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|659029017856
|-
|1.96%
|}
|-
|'''17'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|288
|-
|95.13%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4641
|-
|34.69%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|74273
|-
|25.63%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1188385
|-
|11.20%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|19014177
|-
|16.92%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|304226849
|-
|6.07%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|4867629601
|-
|4.54%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|77882073633
|-
|4.33%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1246113178145
|-
|1.22%
|}
|-
|'''18'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|323
|-
|95.04%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|5526
|-
|29.31%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|93960
|-
|25.32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|1597338
|-
|10.75%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|27154764
|-
|14.81%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|461631006
|-
|5.74%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7847727120
|-
|4.11%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|133411361058
|-
|4.32%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2267993138004
|-
|0.73%
|}
|-
|'''19'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|360
|-
|93.88%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|6517
|-
|25.13%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|117325
|-
|20.43%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2111869
|-
|10.49%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|38013661
|-
|10.58%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|684245917
|-
|5.71%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|12316426525
|-
|4.06%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|221695677469
|-
|3.99%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|3990522194461
|-
|0.45%
|}
|-
|'''20'''
| align="center" |
{| border="2" style="background:#ABCDEF;"
|399
|-
|95.48%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|7620
|-
|25.69%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|144800
|-
|24.13%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|2751220
|-
|10.24%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|52273200
|-
|17.11%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|993190820
|-
|5.6%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|18870625600
|-
|4.04%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|358541886420
|-
|3.61%
|}
| align="center" |
{| border="2" style="background:#ABCDEF;"
|6812295842000
|-
|1.14%
|}
|}
</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: red; text-align: center;" | * || The Moore bound.
|-
|style="background-color: #ABCDEF; text-align: center;" | * || Upper bound introduced by A. Hoffman, R. Singleton, Bannai, E. and Ito, T.
|-
|style="background-color: #999900; text-align: center;" | * || Upper bound introduced by Leif Jorgensen.
|-
|style="background-color: #336666; text-align: center;" | * || Optimal graphs found by Buset and by Molodtsov.
|-
|style="background-color: #ffff00; text-align: center;" | * || Graphs shown optimal.
|-
|}
</center>

==References==
* Abas M., "Large Networks of Diameter Two Based on Cayley Graphs" in "Cybernetics and Mathematics Applications in Intelligent Systems, Advances in Intelligent Systems and Computing 574", (2017), Pages 225-233, [https://arxiv.org/pdf/1509.00842.pdf, PDF version]
* Bannai, E.; Ito, T. (1981), "Regular graphs with excess one", Discrete Mathematics 37:147-158, doi:10.1016/0012-365X(81)90215-6.
* Buset, D. (2000), "Maximal cubic graphs with diameter 4", Discrete Applied Mathematics 101 (1-3): 53-61, doi:10.1016/S0166-218X(99)00204-8.
* J. Dinneen, Michael; Hafner, P. R. (1994), "New Results for the Degree/Diameter Problem", Networks 24 (7): 359–367, [http://arxiv.org/PS_cache/math/pdf/9504/9504214v1.pdf PDF version].
* Elspas, B. (1964), "Topological constraints on interconnection-limited logic", Proceedings of IEEE Fifth Symposium on Switching Circuit Theory and Logical Design S-164: 133--147.
* Erdös P; Fajtlowicz, S.; Hoffman A. J. (1980), "Maximum degree in graphs of diameter 2", Networks 10: 87-90.
* Hoffman, A. J.; Singleton, R. R. (1960), "Moore graphs with diameter 2 and 3", IBM Journal of Research and Development 5 (4): 497–504, MR0140437, [http://www.research.ibm.com/journal/rd/045/ibmrd0405H.pdf PDF version].
* L. K. Jorgensen (1992), "Diameters of cubic graphs", Discrete Applied Mathematics 37/38: 347-351, doi:10.1016/0166-218X(92)90144-Y.
* L. K. Jorgensen (1993), "Nonexistence of certain cubic graphs with small diameters", Discrete Mathematics 114:265-273, doi:10.1016/0012-365X(93)90371-Y.
* Kurosawa, K.; Tsujii, S. (1981), "Considerations on diameter of communication networks", Electronics and Communications in Japan 64A (4): 37-45.
* Loz, E.; Širáň, J. (2008), "New record graphs in the degree-diameter problem", Australasian Journal of Combinatorics 41: 63–80.
* Loz, E.; Pineda-Villavicencio, G. (2010), "New benchmarks for large scale networks with given maximum degree and diameter", The Computer Journal, The British Computer Society, Oxford University Press.
* McKay, B. D.; Miller, M.; Širáň, J. (1998), "A note on large graphs of diameter two and given maximum degree", Journal of Combinatorial Theory Series B 74 (4): 110–118.
* Miller, M; Nguyen, M.; Pineda-Villavicencio, G. (accepted in September 2008), "On the nonexistence of graphs of diameter 2 and defect 2", Journal of Combinatorial Mathematics and Combinatorial Computing.
* Miller, M.; Simanjuntak, R. (2008), "Graphs of order two less than the Moore bound", Discrete Mathematics 308 (13): 2810-2821, doi:10.1016/j.disc.2006.06.045.
* Miller, M.; Širáň, J. (2005), "Moore graphs and beyond: A survey of the degree/diameter problem", Electronic Journal of Combinatorics Dynamic survey D, [http://www.combinatorics.org/Surveys/ds14.pdf PDF version].
* Molodtsov, S. G. (2006), "Largest Graphs of Diameter 2 and Maximum Degree 6", Lecture Notes in Computer Science 4123: 853-857.
* Pineda-Villavicencio, G.; Miller, M. (2008), "On graphs of maximum degree 3 and defect 4", Journal of Combinatorial Mathematics and Combinatorial Computing 65: 25-31.
* Pineda-Villavicencio, G.; Miller, M., "Complete characterization of graphs of maximum degree 3 and defect at most 4", submitted.
* Pineda-Villavicencio, G.; Gómez, J.; Miller, M.; Pérez-Rosés, H., "New Largest Known Graphs of Diameter 6", Networks, to appear, doi:10.1002/net.20269. See also Electronic Notes in Discrete Mathematics 24: 153–160, 2006.
* Pineda-Villavicencio, G.; Miller, M. (Oct 2006), "On Graphs of Maximum Degree 5, Diameter D and Defect 2", Proceedings of MEMICS 2006, Second Doctoral Workshop on Mathematical and Engineering Methods in Computer Science: 182--189, Mikulov, Czech Republic.
* Brown, W. G. (1966) On graphs that do not contain a Thomsen graph. Canadian Mathematical Bulletin, 9, 281 - 285.

==External links==
* [http://www-mat.upc.es/grup_de_grafs/ Degree Diameter] online table.
* [http://www.eyal.com.au/wiki/The_Degree/Diameter_Problem Eyal Loz's] Degree-Diameter problem page.
* [http://isu.indstate.edu/ge/DD/index.html Geoffrey Exoo's] Degree-Diameter record graphs page.

[[Category:The Degree/Diameter Problem]]