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http://combinatoricswiki.org/index.php?title=The_Degree_Diameter_Problem_for_Bipartite_Graphs&diff=649
The Degree Diameter Problem for Bipartite Graphs
2022-02-18T05:55:28Z
<p>Guillermo: </p>
<hr />
<div>==Citation==<br />
<br />
If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.<br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
If you are using a specific page in combinatoricsWiki, say the "The degree-diameter problem for bipartite graphs" page, then it would be better to cite the page as follows. <br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). The degree-diameter problem for bipartite graphs, Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
<br />
==Introduction==<br />
The '''degree/diameter problem for bipartite graphs''' can be stated as follows:<br />
<br />
''Given natural numbers ''d'' and ''k'', find the largest possible number ''N<sup>b</sup>(d,k)'' of vertices in a bipartite graph of maximum degree ''d'' and diameter ''k''.''<br />
<br />
An upper bound for ''N<sup>b</sup>(d,k)'' is given by the so-called ''bipartite Moore bound'' ''M<sup>b</sup>(d,k)=2((d-1)<sup>k</sup>-2)(d-2)<sup>-1</sup>''. Bipartite ''(d,k)''-graphs whose order attains the bipartite Moore bound are called ''bipartite Moore graphs''.<br />
<br />
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.<br />
<br />
Therefore, in attempting to settle the values of ''N<sup>b</sup>(d,k)'', research activities in this problem have follow the following two directions:<br />
<br />
*Increasing the lower bounds for ''N<sup>b</sup>(d,k)'' by constructing ever larger graphs.<br />
<br />
* Lowering and/or setting upper bounds for ''N<sup>b</sup>(d,k)'' by proving the non-existence of graphs<br />
whose order is close to the bipartite Moore bounds ''M<sup>b</sup>(d,k)=2((d-1)<sup>k</sup>-1)(d-2)<sup>-1</sup>''.<br />
<br />
==Increasing the lower bounds for ''N<sup>b</sup>(d,k)''==<br />
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 G&oacute;mez-Mart&iacute;) provided some large bipartite graphs by using graph compounding, the concept of partial Cayley graph, and other techniques. <br />
<br />
Now, with the release of this online table (see below), we expect to stimulate further research on this area.<br />
<br />
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&nbsp;≤&nbsp;''d''&nbsp;≤&nbsp;16 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 3&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;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.<br />
<br />
===Table of the orders of the largest known bipartite graphs===<br />
<br />
<center> <br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
| '''<math>d</math>\<math>k</math>'''|| '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
| '''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<br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|- <br />
| '''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 <br />
|- <br />
| '''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 <br />
|- <br />
| '''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 <br />
|- <br />
| '''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 <br />
|- 97 386 380<br />
| '''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<br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
|}<br />
</center><br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: #bbffff; text-align: center;" | * || Bipartite Moore graphs (optimal).<br />
|-<br />
|style="background-color: #CC6600; text-align: center;" | * || Graph duplications found by C. Delorme and G. Farhi.<br />
|-<br />
|style="background-color: #66ff66; text-align: center;" | * || Graphs found by C. Delorme, J. Gómez, and J. J. Quisquater.<br />
|-<br />
|style="background-color: #0066CC; text-align: center;" | * || Optimal graph found by R. Bar-Yehuda and T. Etzion and by J. Bond and C. Delorme.<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #ff0000; text-align: center;" | * || Graphs found independently by Paul Hafner and by Eyal Loz. <br />
|-<br />
|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.<br />
|-<br />
|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. <br />
|-<br />
|style="background-color: pink; text-align: center;" | * || Graphs found by G. Araujo and N. López. <br />
|}<br />
</center><br />
<br />
==Lowering and/or setting upper bounds for ''N<sup>b</sup>(d,k)''== <br />
<br />
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 ''M<sup>b</sup>(d,k)-&delta;'' for small ''&delta;''. The parameter ''&delta;'' is called the defect. Such graphs are called bipartite ''(d,k,-&delta;)''-graphs.<br />
<br />
The bipartite ''(d,k,-2;)''-graphs constitute the first interesting family of graphs to be studied. When ''d&ge;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&ge;3'' and ''k&ge;3'' only two such graphs are known; a unique bipartite ''(3, 3,-2)''-graph and a unique bipartite ''(4, 3,-2)''-graph.<br />
<br />
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&ge;3'' and ''k&ge;3'', and the non-existence of bipartite ''(d,k,-2;)''-graphs for ''d&ge;3'' and ''k&ge;4''. <br />
<br />
===Lowest known upper bounds and the percentage of the order of the largest known bipartite graphs===<br />
<br />
<center><br />
{| border="1"<br />
| '''<math>d</math>\<math>k</math>'''|| '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
|'''3'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''14'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''30'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|'''56'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''126'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|248<br />
|-<br />
|67.74%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|504<br />
|-<br />
|50.79%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|1016<br />
|-<br />
|49.80%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|2040<br />
|-<br />
|39.21%<br />
|}<br />
|-<br />
|'''4'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''26'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''80'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|236<br />
|-<br />
|67.79%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''728'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|2180<br />
|-<br />
|38.53%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|6554<br />
|-<br />
|33.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|19676<br />
|-<br />
|25.25%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|59042<br />
|-<br />
|19.89%<br />
|}<br />
|-<br />
|'''5'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''42'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''170'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|676<br />
|-<br />
|49.70%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''2730'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|10916<br />
|-<br />
|28.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|43684<br />
|-<br />
|21.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|174756<br />
|-<br />
|15.98%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|699044<br />
|-<br />
|12.88%<br />
|}<br />
|-<br />
|'''6'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''62'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''312'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|1556<br />
|-<br />
|43.95%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''7812'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|39056<br />
|-<br />
|21.27%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|195306<br />
|-<br />
|15.25%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|976556<br />
|-<br />
|12.01%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|4882806<br />
|-<br />
|9.25%<br />
|}<br />
|-<br />
|'''7'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|'''80'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|518<br />
|-<br />
|66.79%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|3106<br />
|-<br />
|36.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|18662<br />
|-<br />
|48.18%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|111968<br />
|-<br />
|20.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|671840<br />
|-<br />
|12.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|4031072<br />
|-<br />
|9.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|24186464<br />
|-<br />
|8.21%<br />
|}<br />
|-<br />
|'''8'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''114'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''800'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|5596<br />
|-<br />
|30.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''39216'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|274508<br />
|-<br />
|14.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|1921594<br />
|-<br />
|10.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|13451196<br />
|-<br />
|8.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|94158410<br />
|-<br />
|7.35%<br />
|}<br />
|-<br />
|'''9'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''146'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''1170'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|9356<br />
|-<br />
|26.67%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''74898'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|599180<br />
|-<br />
|19.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|4793484<br />
|-<br />
|9.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|38347916<br />
|-<br />
|7.72%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|306783372<br />
|-<br />
|6.52%<br />
|}<br />
|-<br />
|'''10'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''182'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''1640'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|14756<br />
|-<br />
|27.10%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''132860'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|1195736<br />
|-<br />
|18.79%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|10761674<br />
|-<br />
|10.93%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|96855116<br />
|-<br />
|7.28%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|871696094<br />
|-<br />
|5.77%<br />
|}<br />
|-<br />
|'''11'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(100,100,255);" <br />
|220<br />
|-<br />
|86.36%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|2222<br />
|-<br />
|78.03%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|22216<br />
|-<br />
|26.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|222222<br />
|-<br />
|64.10%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|2222216<br />
|-<br />
|17.93%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|22222216<br />
|-<br />
|10.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|222222216<br />
|-<br />
|6.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|2222222216<br />
|-<br />
|5.87%<br />
|}<br />
|-<br />
|'''12'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''266'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''2928'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|32204<br />
|-<br />
|25.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''354312'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|3897428<br />
|-<br />
|17.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|42871770<br />
|-<br />
|10.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|471589532<br />
|-<br />
|6.36%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|5187484914<br />
|-<br />
|5.79%<br />
|}<br />
|-<br />
|'''13'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|314<br />
|-<br />
|85.98%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|3770<br />
|-<br />
|81.27%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|45236<br />
|-<br />
|25.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|542906<br />
|-<br />
|68.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|6514868<br />
|-<br />
|16.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|78178484<br />
|-<br />
|6.79%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|938141876<br />
|-<br />
|5.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|11257702580<br />
|-<br />
|5.48%<br />
|}<br />
|-<br />
|'''14'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''366'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''4760'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|61876<br />
|-<br />
|23.85%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|'''804468'''<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|10458080<br />
|-<br />
|16.93%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|135955114<br />
|-<br />
|10.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|1767416556<br />
|-<br />
|5.38%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|22976415302<br />
|-<br />
|5.28%<br />
|}<br />
|-<br />
|'''15'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|422<br />
|-<br />
|87.67%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|5910<br />
|-<br />
|83.68%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|82736<br />
|-<br />
|24.77%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|1158390<br />
|-<br />
|72.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|16217456<br />
|-<br />
|15.29%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|227044464<br />
|-<br />
|0%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|3178622576<br />
|-<br />
|5.28%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|44500716144<br />
|-<br />
|5.17%<br />
|}<br />
|-<br />
|'''16'''<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|482<br />
|-<br />
|81.74%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|7232<br />
|-<br />
|70.99%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|108476<br />
|-<br />
|25.16%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(40,40,255);" <br />
|1627232<br />
|-<br />
|54.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|24408476<br />
|-<br />
|16.47%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|366127226<br />
|-<br />
|9.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|5491908476<br />
|-<br />
|5.26%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:rgb(210,210,255);" <br />
|82378627226<br />
|-<br />
|5%<br />
|}<br />
|}<br />
</center><br />
<br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: rgb(40,40,255); text-align: center;" | * || Moore bound.<br />
|-<br />
|style="background-color: rgb(100,100,255); text-align: center;" | * || Moore bound minus 2.<br />
|-<br />
|style="background-color: rgb(210,210,255); text-align: center;" | * || Moore bound minus 6. <br />
|-<br />
|}<br />
</center><br />
<br />
==References==<br />
* Conder, M.; Nedela, R. (2006), "A more detailed classification of symmetric cubic graphs", preprint.<br />
* Bar-Yehuda, R.; Etzion, T. (1992), "Connections between two cycles - a new design of dense processor interconnection networks", Discrete Applied Mathematics 37-38.<br />
* Bond, J.; Delorme, C. (1988), "New large bipartite graphs with given degree and diameter", Ars Combinatoria 25C: 123-132.<br />
* 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.<br />
* Delorme, C. (1985), "Grands graphes de degr&eacute; et diam&egrave;tre donn&eacute;s", European Journal of Combinatorics 6: 291-302.<br />
* Delorme, C. (1985), "Large bipartite graphs with given degree and diameter", Journal of Graph Theory 8: 325-334.<br />
* Delorme, C.; Farhi, G. (1984), "Large graphs with given degree and diameter Part I", IEEE Transactions on Computers C-33: 857-860.<br />
* Delorme, C.; G&oacute;mez (2002), "Some new large compound graphs", European Journal of Combinatorics 23 (5): 539-547, doi:10.1006/eujc.2002.0581.<br />
* Delorme, C.; Gómez, J.; Quisquater, J. J., "On large bipartite graphs", submitted.<br />
* 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.<br />
* 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.<br />
*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.<br />
* 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].<br />
<br />
==External links==<br />
* [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''.<br />
<br />
<br />
<br />
<br />
<br />
[[Category:The Degree/Diameter Problem]]</div>
Guillermo
http://combinatoricswiki.org/index.php?title=The_Degree_Diameter_Problem_for_Planar_Graphs&diff=648
The Degree Diameter Problem for Planar Graphs
2022-02-18T05:54:19Z
<p>Guillermo: </p>
<hr />
<div>==Citation==<br />
<br />
If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.<br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
If you are using a specific page in combinatoricsWiki, say the "The degree-diameter problem for planar graphs" page, then it would be better to cite the page as follows. <br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). The degree-diameter problem for planar graphs, Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
<br />
==Table of the orders of the largest known regular planar graphs for the undirected degree diameter problem==<br />
<br />
<center> <br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8'''<br />
|-<br />
| '''3''' || style="background-color: blue; text-align: center;" |6 || style="background-color: blue; text-align: center;" |12 ||style="background-color: blue; text-align: center;" | 18 || style="background-color: blue; text-align: center;" |28 ||style="background-color: brown; text-align: center;" | 36 || style="background-color: brown; text-align: center;" |52 ||style="background-color: brown; text-align: center;" | 76<br />
|-<br />
| '''4''' ||style="background-color: blue; text-align: center;" | 9 || style="background-color: blue; text-align: center;" |16 || style="background-color: yellow; text-align: center;" |27 || style="background-color: brown; text-align: center;" |44 ||style="background-color: brown; text-align: center;" | 81 || style="background-color: brown; text-align: center;" |134 ||style="background-color: brown; text-align: center;" | 243<br />
|-<br />
| '''5''' || NA ||style="background-color: blue; text-align: center;" | '''16''' || style="background-color: yellow; text-align: center;" |28 || style="background-color: brown; text-align: center;" |62 || style="background-color: brown; text-align: center;" |124 ||style="background-color: brown; text-align: center;" | 254 ||style="background-color: brown; text-align: center;" | 500<br />
|-<br />
|}<br />
</center><br />
<br />
Optimal graphs are marked in bold. The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: yellow; text-align: center;" | * || Graphs found by Preen.<br />
|-<br />
|style="background-color: brown; text-align: center;" | * || Graphs found by Pratt and Friedman.<br />
|}<br />
</center><br />
<br />
==Table of the orders of the largest known planar graphs for the undirected degree diameter problem==<br />
<br />
<center> <br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
| '''3''' || style="background-color: red; text-align: center;" | '''7''' || style="background-color: blue; text-align: center;" | '''12''' || style="background-color: blue; text-align: center;" | 18 || style="background-color: blue; text-align: center;" | 28 || style="background-color: purple; text-align: center;" | 38||style="background-color: red; text-align: center;" | 53||style="background-color: red; text-align: center;" | 77||style="background-color: red; text-align: center;" |109||style="background-color: red; text-align: center;" |157<br />
|-<br />
| '''4''' || style="background-color: green; text-align: center;" | '''9''' || style="background-color: blue; text-align: center;" | 16 || style="background-color: blue; text-align: center;" | 27 || style="background-color: red; text-align: center;" | 44 || style="background-color: blue; text-align: center;" | 81||style="background-color: red; text-align: center;" |134||style="background-color: yellow; text-align: center;" | 243||style="background-color: red; text-align: center;" |404||style="background-color: red; text-align: center;" |728<br />
|-<br />
| '''5''' || style="background-color: green; text-align: center;" | '''10''' || style="background-color: red; text-align: center;" | 19 || style="background-color: purple; text-align: center;" | 39 || style="background-color: red; text-align: center;" | 73 || style="background-color:yellow; text-align: center;" | 158||style="background-color: red; text-align: center;" |289||style="background-color: yellow; text-align: center;" |638||style="background-color: red; text-align: center;" |1153||style="background-color:yellow; text-align: center;" |2558<br />
|-<br />
| '''6''' || style="background-color: green; text-align: center;" | '''11''' || style="background-color: red; text-align: center;" | 24 || style="background-color:yellow; text-align: center;" | 55 || style="background-color: #00ff7f; text-align: center;" | 117 || style="background-color: yellow; text-align: center;" | 280||style="background-color: #00ff7f; text-align: center;" |579||style="background-color:yellow; text-align: center;" |1405||style="background-color: #00ff7f; text-align: center;" |2889||style="background-color:yellow; text-align: center;" |7030<br />
|-<br />
| '''7''' || style="background-color: green; text-align: center;" | '''12''' || style="background-color: red; text-align: center;" | 28 || style="background-color:yellow; text-align: center;" | 74 || style="background-color: #00ff7f; text-align: center;" | 165 || style="background-color: yellow; text-align: center;" | 452||style="background-color: #00ff7f; text-align: center;" | 984||style="background-color:yellow; text-align: center;" |2720||style="background-color: #00ff7f; text-align: center;" |5898||style="background-color:yellow; text-align: center;" |16328<br />
|-<br />
| '''8''' || style="background-color: red; text-align: center;" | '''13'''|| style="background-color: red; text-align: center;" | 33 || style="background-color:yellow; text-align: center;" | 97 || style="background-color: #00ff7f; text-align: center;" | 228 || style="background-color: yellow; text-align: center;" |685||style="background-color: #00ff7f; text-align: center;" | 1590||style="background-color:yellow; text-align: center;" |4901||style="background-color: #00ff7f; text-align: center;" |11124||style="background-color:yellow; text-align: center;" |33613<br />
|-<br />
| '''9''' || style="background-color: red; text-align: center;" | '''14''' || style="background-color: red; text-align: center;" | 37 || style="background-color:yellow; text-align: center;" | 122 || style="background-color: #00ff7f; text-align: center;" | 293 || style="background-color: yellow; text-align: center;" |986||style="background-color: #00ff7f; text-align: center;" | 2338||style="background-color:yellow; text-align: center;" |7898||style="background-color: #00ff7f; text-align: center;" |18698||style="background-color:yellow; text-align: center;" |63194<br />
|-<br />
| '''10''' || style="background-color: red; text-align: center;" | '''16''' || style="background-color: red; text-align: center;" | 42 || style="background-color: yellow; text-align: center;" | 151 || style="background-color: #00ff7f; text-align: center;" | 375 || style="background-color: yellow; text-align: center;" | 1366||style="background-color: #00ff7f; text-align: center;" | 3369||style="background-color:yellow; text-align: center;" |12301||style="background-color: #00ff7f; text-align: center;" |30315||style="background-color:yellow; text-align: center;" |110716<br />
|-<br />
|}<br />
</center><br />
<br />
Optimal graphs are marked in bold. The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: blue; text-align: center;" | * || Graphs of unknown author.<br />
|-<br />
|style="background-color: red; text-align: center;" | * || Graphs found by Fellows, Hell, and Seyffarth. Details are available in a paper by the authors.<br />
|-<br />
|style="background-color: green; text-align: center;" | * || Graphs found by Yang, Lin, and Dai. Details are available in a paper by the authors.<br />
|-<br />
|style="background-color: purple; text-align: center;" | * || Graphs found by Geoffrey Exoo.<br />
|-<br />
|style="background-color: yellow; text-align: center;" | * || Graphs found by S. A. Tishchenko. Details are available in a paper by the author.<br />
|-<br />
|style="background-color: #00ff7f; text-align: center;" | * || Graphs found by R. Feria-Purón and G. Pineda-Villavicencio. Details are available in a paper by the authors.<br />
|}<br />
</center><br />
<br />
==References==<br />
<br />
* Fellows, M.; Hell, P.; Seyffarth, K. (1998), "Constructions of large planar networks with given degree and diameter", Networks 32: 275-281.<br />
<br />
* Feria-Purón, R.; Pineda-Villavicencio, G. (2013), "Constructions of large graphs on surfaces", preprint, [http://arxiv.org/pdf/1302.1648v1.pdf PDF version]. <br />
<br />
* Tishchenko, S. A. (2012), "Maximum size of a planar graph with given degree and even diameter", European Journal of Combinatorics 33: 380-396.<br />
<br />
* Yang, Y.; Lin, J.; Dai, Y. (2002), "Largest planar graphs and largest maximal planar graphs of diameter two", Journal of Computational and Applied Mathematics 144:349-358.<br />
<br />
==External links==<br />
<br />
*[http://faculty.cbu.ca/jpreen/degdiam.html Tables of the largest known planar graphs maintained by James Preen]<br />
<br />
<br />
<br />
<br />
<br />
[[Category:The Degree/Diameter Problem]]</div>
Guillermo
http://combinatoricswiki.org/index.php?title=The_Degree_Diameter_Problem_for_Cayley_Graphs&diff=647
The Degree Diameter Problem for Cayley Graphs
2022-02-18T05:52:38Z
<p>Guillermo: </p>
<hr />
<div>==Citation==<br />
<br />
If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.<br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
If you are using a specific page in combinatoricsWiki, say the "The degree-diameter problem for Cayley graphs" page, then it would be better to cite the page as follows. <br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). The degree-diameter problem for Cayley graphs, Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
<br />
==Introduction==<br />
<br />
The '''degree/diameter problem for Cayley graphs''' can be stated as follows:<br />
<br />
''Given natural numbers ''d'' and ''k'', find the largest possible number ''N<sup>c</sup>(d,k)'' of vertices in a [http://mathworld.wolfram.com/CayleyGraph.html Cayley graph] of maximum degree ''d'' and diameter ''k''.''<br />
<br />
There are no better upper bounds for ''N<sup>c</sup>(d,k)'' than the very general ''Moore bounds'' ''M(d,k)=(d(d-1)<sup>k</sup>-2)(d-2)<sup>-1</sup>''. As an interesting fact, no [http://en.wikipedia.org/wiki/Moore_graph Moore graph] (graph whose order attains the Moore bound) is a Cayley graph. Indeed, neither the [http://en.wikipedia.org/wiki/Petersen_graph Petersen graph] nor the [http://en.wikipedia.org/wiki/Hoffman-Singleton_graph Hoffman-Singleton graph] are Cayley, and the possible Moore graph of degree 57 and diameter 2 is not even [http://en.wikipedia.org/wiki/Vertex-transitive_graph vertex-transitive]. <br />
<br />
Therefore, in attempting to settle the values of ''N<sup>c</sup>(d,k)'', research activities in this problem follow the next two directions:<br />
<br />
* Increasing the lower bounds for ''N<sup>c</sup>(d,k)'' by constructing ever larger graphs.<br />
<br />
* Lowering and/or setting upper bounds for ''N<sup>c</sup>(d,k)'' by proving the non-existence of graphs whose order is close to the Moore bounds ''M(d,k)''.<br />
<br />
Finding an upper bound for the general [http://mathworld.wolfram.com/Non-AbelianGroup.html non-abelian] case is still an open problem, while such an upper bound for the abelian case is already known.<br />
<br />
==Increasing the lower bounds for N<sup>c</sup>(d,k)==<br />
<br />
The current largest lower bounds (of order close to d<sup>2</sup>/2) for Cayley graphs of diameter ''k''=2 and an infinite set of values for the degree ''d''>20, is given by Šiagiová and Širáň. Some lower bounds for trivalent graphs of diameter ''d''>10 are given by Curtin.<br />
<br />
===Table of the orders of the largest known Cayley graphs for the undirected degree diameter problem===<br />
<br />
Below is the table of the largest known [http://en.wikipedia.org/wiki/Cayley_graph Cayley] graphs (as of September 2009) in the undirected [[The Degree/Diameter Problem | degree diameter problem]] for Cayley graphs of [http://en.wikipedia.org/wiki/Degree_(graph_theory) degree] at most 3&nbsp;≤&nbsp;''d''&nbsp;≤&nbsp;20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;10. This table represents the best lower bounds known at present on the order of Cayley ''(d,k)''-graphs. All optimal graphs are marked in bold. All Cayley graphs of order up to 33 are isomorphic to graphs in the lists available [http://people.csse.uwa.edu.au/gordon/remote/cayley/index.html here], and some of the trivalent Cayley graphs of order up to 1000 are available [http://people.csse.uwa.edu.au/gordon/remote/cubcay here], but no information was given in these lists regarding diameter.<br />
<br />
<center> <br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
| '''3''' || '''8''' || style="background-color: #ccff33;" | '''14''' || style="background-color: #ccff33;" | '''24''' ||style="background-color: #006600;" | '''60''' || style="background-color: #ccff33;" | '''72''' ||style="background-color: #ccff33;" | '''168''' ||style="background-color: #ff6600;" | '''300'''||style="background-color: #ff6600;" | 506||style="background-color: #ff9999;" | 882<br />
|-<br />
<br />
| '''4''' ||style="background-color: #ccff33;" | '''13''' ||style="background-color: #ccff33;" | '''30''' ||style="background-color: #339900;" | '''84''' || style="background-color: #ccff33;" | 216 ||style="background-color: #ff6600;" | 513||style="background-color: #FF0066;" | 1 155 ||style="background-color: #bbffff;" | 3 080 ||style="background-color: #FF0066;" | 7 550 ||style="background-color: #bbffff;" | 17 604 <br />
|-<br />
| '''5''' ||style="background-color: #ccff33;" | '''18''' ||style="background-color: #ccff33;" | '''60''' ||style="background-color: #bbffff;" | 210 ||style="background-color: #bbffff;" | 546||style="background-color: #ff6600;" | 1 640 ||style="background-color: #bbffff;" | 5 500 ||style="background-color: #bbffff;" | 16 965 ||style="background-color: #FF9900;" | 57 840 ||style="background-color: #FF9900;" | 187 056 <br />
|-<br />
| '''6''' || style="background-color: #ccff33;" |'''32'''||style="background-color: #CCCCFF;" | '''108''' ||style="background-color: #bbffff;" | 375 ||style="background-color: #bbffff;" | 1 395 ||style="background-color: #ff6600;" | 5 115 ||style="background-color: #FF9900;" | 19 383 ||style="background-color: #FF9900;" | 76 461 ||style="background-color: #FF9900;" | 307 845 ||style="background-color: #FF9900;" | 1 253 615 <br />
|-<br />
| '''7''' ||style="background-color: #ccff33;" |'''36''' ||style="background-color: #ccff33;" |'''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 <br />
|-<br />
| '''8''' ||style="background-color: #ccff33;" |'''48'''||style="background-color: #993300;" | 253 ||style="background-color: #FF9900;" | 1 100 ||style="background-color: #FF9900;" | 5 060 ||style="background-color: #ff6600;" | 23 991 ||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 <br />
|-<br />
| '''9''' ||style="background-color: #ccff33;" | '''60''' || style="background-color: #ff6600;" | 294 ||style="background-color: #FF9900;" | 1 550 ||style="background-color: #FF9900;" | 8 200 ||style="background-color: #ff6600;" | 45 612 ||style="background-color: #FF9900;" | 279 616 ||style="background-color: #FF9900;" | 1 686 600 ||style="background-color: #FF9900;" | 12 123 288 ||style="background-color: #FF9900;" | 65 866 350 <br />
|-<br />
| '''10''' ||style="background-color: #ccff33;" | '''72''' || style="background-color: #ff6600;" | 406 ||style="background-color: #FF9900;" | 2 286 ||style="background-color: #FF9900;" | 13 140 ||style="background-color: #ff6600;" | 81 235 ||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 <br />
|-<br />
| '''11''' ||style="background-color: #ccff33;" | '''84''' ||style="background-color: #ff6600;" | 486 || style="background-color: #ff6600;" | 2 860 ||style="background-color: #FF9900;" | 19 500 ||style="background-color: #ff6600;" | 139 446 ||style="background-color: #FF9900;" | 1 001 268 ||style="background-color: #FF9900;" | 7 442 328 || style="background-color: #FF9900;" | 72 933 102 ||style="background-color: #ff6600;" | 500 605 110<br />
|-<br />
| '''12''' ||style="background-color: #ccff33;" | '''96''' ||style="background-color: #ff6600;" | 605 ||style="background-color: #ff6600;" | 3 775 ||style="background-color: #FF9900;" | 29 470||style="background-color: #ff6600;" | 229 087 ||style="background-color: #FF9900;" | 1 999 500 ||style="background-color: #FF9900;" | 15 924 326 ||style="background-color: #FF9900;" | 158 158 875 ||style="background-color: #ff6600;" | 1 225 374 192<br />
|-<br />
| '''13''' ||style="background-color: #ff6600;" | 112 ||style="background-color: #ff6600;" | 680 ||style="background-color: #ff6600;" | 4 788 ||style="background-color: #FF9900;" |40 260 ||style="background-color: #ff6600;" | 347 126 ||style="background-color: #FF9900;" | 3 322 080 ||style="background-color: #FF9900;" | 29 927 790 ||style="background-color: #ff6600;" | 233 660 788 ||style="background-color: #ff6600;" | 2 129 329 324<br />
|-<br />
| '''14''' ||style="background-color: #ff6600;" | 128 ||style="background-color: #ff6600;" | 873 ||style="background-color: #ff6600;" | 6 510 ||style="background-color: #FF9900;" | 57 837 ||style="background-color: #ff6600;" | 530 448 ||style="background-color: #ff6600;" | 5 600 532 ||style="background-color: #ff6600;" | 50 128 239 ||style="background-color: #ff6600;" | 579 328 377 ||style="background-color: #FF9900;" | 7 041 746 081<br />
|-<br />
| '''15''' ||style="background-color: #ff6600;" | 144 || style="background-color: #ff6600;" | 972 || style="background-color: #ff6600;" | 7 956 ||style="background-color: #FF9900;" | 76 518 || style="background-color: #ff6600;" | 787 116 ||style="background-color: #FF9900;" | 8 599 986 ||style="background-color: #ff6600;" | 88 256 520 ||style="background-color: #ff6600;" | 1 005 263 436 ||style="background-color: #FF9900;" | 10 012 349 898<br />
|-<br />
| '''16''' ||style="background-color: yellow;" | 200 ||style="background-color: #ff6600;" | 1 155 ||style="background-color: #ff6600;" | 9 576 ||style="background-color: #ff6600;" | 100 650 || style="background-color: #ff6600;" | 1 125 264 || style="background-color: #ff6600;" | 12 500 082 ||style="background-color: #ff6600;" | 135 340 551 ||style="background-color: #ff6600;" | 1 995 790 371 ||style="background-color: #FF9900;" | 12 951 451 931<br />
|-<br />
| '''17''' ||style="background-color: yellow;" | 200 ||style="background-color: #ff6600;" | 1 260 ||style="background-color: #ff6600;" | 12 090 ||style="background-color: #ffff66;" | 133 144 || style="background-color: #ff6600;" | 1 609 830 || style="background-color: #ffff66;" | 18 495 162 ||style="background-color: #ffff66;" | 220 990 700 ||style="background-color: #ffff66;" | 3 372 648 954 ||style="background-color: #666666;" | <br />
<br />
|-<br />
| '''18''' ||style="background-color: yellow;" | 200 ||style="background-color: #ff6600;" | 1 510 ||style="background-color: #ff6600;" | 15 026 ||style="background-color: #ffff66;" | 171 828 || style="background-color: #ff6600;" | 2 193 321 || style="background-color: #ffff66;" | 26 515 120 ||style="background-color: #ffff66;" | 323 037 476 ||style="background-color: #ffff66;" | 5 768 971 167 ||style="background-color: #666666;" | <br />
<br />
|-<br />
| '''19''' ||style="background-color: #ff6600;" | 200 ||style="background-color: #ffff66;" | 1 638 ||style="background-color: #ff6600;" | 17 658 ||style="background-color: #ffff66;" | 221 676 || style="background-color: #ff6600;" | 3 030 544 || style="background-color: #ffff66;" | 39 123 116 ||style="background-color: #ffff66;" | 501 001 000 ||style="background-color: #ffff66;" | 8 855 580 344 ||style="background-color: #666666;" | <br />
<br />
|-<br />
| '''20''' ||style="background-color: #ff6600;" | 210 ||style="background-color: #ffff66;" | 1 958 ||style="background-color: #ff6600;" | 21 333 ||style="background-color: #ffff66;" | 281 820 || style="background-color: #ff6600;" | 4 040 218 || style="background-color: #ffff66;" | 55 625 185 ||style="background-color: #ffff66;" | 762 374 779 ||style="background-color: #ffff66;" | 12 951 451 931 ||style="background-color: #666666;" | <br />
|}<br />
</center><br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #993300; text-align: center;" | * || Graph found by Mitjana M. and Francesc Comellas. This graph was also found independently by Michael Sampels.<br />
|-<br />
|style="background-color: #CCCCFF; text-align: center;" | * || Graph found by Wohlmuth, and shown to be optimal by Marston Conder.<br />
|-<br />
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by Michael Sampels.<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #339900; text-align: center;" | * || Optimal graph found by Marston Conder. This graph was also found independently by Eyal Loz.<br />
|-<br />
|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.<br />
|-<br />
|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.<br />
|-<br />
|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. <br />
|-<br />
|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.<br />
|-<br />
|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''.<br />
|-<br />
|style="background-color: yellow; text-align: center;" | * || Graphs found by Marcel Abas.<br />
|}<br />
</center><br />
<br />
==Lowering and/or setting upper bounds for N<sup>c</sup>(d,k)==<br />
<br />
In attempting to set the values for ''N<sup>c</sup>(d,k)'', most research efforts have been directed at the abelian case.<br />
<br />
'''The abelian case'''<br />
<br />
When the group considered is abelian, a Moore-like bound was given by Stanton and Cowan, and is asymptotically equivalent to (2''k'')<sup>''d''/2</sup>(''d''/2)!<sup>-1</sup>, where (''d''/2) is the number of generators from the group. Dougherty and Faber gave a list of optimal graphs for both the directed and undirected abelian Cayley case.<br />
<br />
'''The non-abelian case'''<br />
<br />
As said above, to set a Moore-like bound for the non-abelian case remains an open problem.<br />
<br />
===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===<br />
<br />
<center><br />
{| border="1"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
|'''3'''<br />
| align="center" |<br />
{| border="2"<br />
|8<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|14<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|24<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|60<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|72<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|168<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ff9999;" <br />
|300<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1532<br />
|-<br />
|33.02%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3068<br />
|-<br />
|28.75%<br />
|}<br />
|-<br />
|'''4'''<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|13<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|30<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|84<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484<br />
|-<br />
|42.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1456<br />
|-<br />
|35.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4372<br />
|-<br />
|26.41%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13120<br />
|-<br />
|23.47%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|39364<br />
|-<br />
|19.17%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|118096<br />
|-<br />
|14.90%<br />
|}<br />
|-<br />
|'''5'''<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|18<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|60<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|424<br />
|-<br />
|49.52%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1704<br />
|-<br />
|32.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6824<br />
|-<br />
|24.03%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27304<br />
|-<br />
|20.14%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|109224<br />
|-<br />
|15.53%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|436904<br />
|-<br />
|13.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1747624<br />
|-<br />
|10.70%<br />
|}<br />
|-<br />
|'''6'''<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|32<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|108<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|936<br />
|-<br />
|40.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4686<br />
|-<br />
|29.76%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23436<br />
|-<br />
|21.82%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117186<br />
|-<br />
|16.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|585936<br />
|-<br />
|13.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929686<br />
|-<br />
|10.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14648436<br />
|-<br />
|8.55%<br />
|}<br />
|-<br />
|'''7'''<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|36<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|168<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1813<br />
|-<br />
|37.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|10885<br />
|-<br />
|25.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|65317<br />
|-<br />
|18.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|391909<br />
|-<br />
|13.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2351461<br />
|-<br />
|10.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14108773<br />
|-<br />
|8.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|84652645<br />
|-<br />
|7.09%<br />
|}<br />
|-<br />
|'''8'''<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|48<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|456<br />
|-<br />
|55.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3200<br />
|-<br />
|34.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|22408<br />
|-<br />
|22.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|156864<br />
|-<br />
|15.29%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1098056<br />
|-<br />
|11.94%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7686400<br />
|-<br />
|9.56%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53804808<br />
|-<br />
|7.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|376633664<br />
|-<br />
|6.61%<br />
|}<br />
|-<br />
|'''9'''<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|60<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|657<br />
|-<br />
|44.74%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5265<br />
|-<br />
|29.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42129<br />
|-<br />
|19.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|337041<br />
|-<br />
|13.53%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2696337<br />
|-<br />
|10.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|21570705<br />
|-<br />
|7.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|172565649<br />
|-<br />
|7.02%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1380525201<br />
|-<br />
|4.77%<br />
|}<br />
|-<br />
|'''10'''<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|72<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|910<br />
|-<br />
|44.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8200<br />
|-<br />
|27.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73810<br />
|-<br />
|17.80%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|664300<br />
|-<br />
|12.22%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5978710<br />
|-<br />
|9.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53808400<br />
|-<br />
|7.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484275610<br />
|-<br />
|5.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4358480500<br />
|-<br />
|4.61%<br />
|}<br />
|-<br />
|'''11'''<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|84<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1221<br />
|-<br />
|39.80%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12221<br />
|-<br />
|23.40%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122221<br />
|-<br />
|15.95%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222221<br />
|-<br />
|11.40%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222221<br />
|-<br />
|8.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122222221<br />
|-<br />
|6.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222222221<br />
|-<br />
|5.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222222221<br />
|-<br />
|4.09%<br />
|}<br />
|-<br />
|'''12'''<br />
| align="center" |<br />
{| border="2" style="background:#ccff33;" <br />
|96<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1596<br />
|-<br />
|37.90%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|17568<br />
|-<br />
|21.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|193260<br />
|-<br />
|15.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2125872<br />
|-<br />
|10.77%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23384604<br />
|-<br />
|8.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|257230656<br />
|-<br />
|6.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2829537228<br />
|-<br />
|5.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|31124909520<br />
|-<br />
|3.93%<br />
|}<br />
|-<br />
|'''13'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|168<br />
|-<br />
|65.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2041<br />
|-<br />
|33.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|24505<br />
|-<br />
|19.53%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|294073<br />
|-<br />
|13.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3528889<br />
|-<br />
|9.83%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42346681<br />
|-<br />
|7.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|508160185<br />
|-<br />
|5.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6097922233<br />
|-<br />
|3.83%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73175066809<br />
|-<br />
|2.90%<br />
|}<br />
|-<br />
|'''14'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195<br />
|-<br />
|65.64%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2562<br />
|-<br />
|34.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|33320<br />
|-<br />
|19.53%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|433174<br />
|-<br />
|13.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5631276<br />
|-<br />
|9.41%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73206602<br />
|-<br />
|7.65%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|951685840<br />
|-<br />
|5.26%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12371915934<br />
|-<br />
|4.68%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160834907156<br />
|-<br />
|4.37%<br />
|}<br />
|-<br />
|'''15'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|224<br />
|-<br />
|64.28%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3165<br />
|-<br />
|30.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|44325<br />
|-<br />
|17.94%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|620565<br />
|-<br />
|12.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8687925<br />
|-<br />
|9.05%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|121630965<br />
|-<br />
|7.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1702833525<br />
|-<br />
|5.18%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23839669365<br />
|-<br />
|4.21%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|333755371125<br />
|-<br />
|2.99%<br />
|}<br />
|-<br />
|'''16'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|255<br />
|-<br />
|60.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3856<br />
|-<br />
|29.95%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|57856<br />
|-<br />
|16.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|867856<br />
|-<br />
|11.59%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13017856<br />
|-<br />
|8.64%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195267856<br />
|-<br />
|6.40%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929017856<br />
|-<br />
|4.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|43935267856<br />
|-<br />
|4.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|659029017856<br />
|-<br />
|1.96%<br />
|}<br />
|-<br />
|'''17'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|288<br />
|-<br />
|58.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4641<br />
|-<br />
|27.14%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|74273<br />
|-<br />
|16.27%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1188385<br />
|-<br />
|11.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|19014177<br />
|-<br />
|8.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|304226849<br />
|-<br />
|6.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4867629601<br />
|-<br />
|4.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|77882073633<br />
|-<br />
|4.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1246113178145<br />
|-<br />
|0%<br />
|}<br />
|-<br />
|'''18'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|323<br />
|-<br />
|59.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5526<br />
|-<br />
|27.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|93960<br />
|-<br />
|15.99%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1597338<br />
|-<br />
|10.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27154764<br />
|-<br />
|8.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|461631006<br />
|-<br />
|5.74%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7847727120<br />
|-<br />
|4.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|133411361058<br />
|-<br />
|4.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2267993138004<br />
|-<br />
|0%<br />
|}<br />
|-<br />
|'''19'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|360<br />
|-<br />
|55.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6517<br />
|-<br />
|25.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117325<br />
|-<br />
|15.05%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2111869<br />
|-<br />
|10.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|38013661<br />
|-<br />
|7.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|684245917<br />
|-<br />
|5.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12316426525<br />
|-<br />
|4.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|221695677469<br />
|-<br />
|3.99%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3990522194461<br />
|-<br />
|0%<br />
|}<br />
|-<br />
|'''20'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|399<br />
|-<br />
|52.36%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7620<br />
|-<br />
|25.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|144800<br />
|-<br />
|14.73%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2751220<br />
|-<br />
|10.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52273200<br />
|-<br />
|7.72%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|993190820<br />
|-<br />
|5.6%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|18870625600<br />
|-<br />
|4.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|358541886420<br />
|-<br />
|3.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6812295842000<br />
|-<br />
|0%<br />
|}<br />
|}<br />
</center><br />
<br />
<br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: #ccff33; text-align: center;" | * || Graphs shown to be optimal by Marston Conder.<br />
|-<br />
|style="background-color: #ff9999; text-align: center;" | * || Graphs shown to be optimal by P. Potočnik, P. Spiga and G. Verret.<br />
|-<br />
|style="background-color: #ABCDEF; text-align: center;" | * || Upper bound introduced by A. Hoffman, R. Singleton, Bannai, E. and Ito, T.<br />
|-<br />
|}<br />
</center><br />
<br />
== Cayley Graphs of Diameter Two ==<br />
<br />
For the special case of diameter 2, Cayley graphs of degree up to 12 are known to be optimal. It is interesting to note that optimal Cayley graphs are smaller than the Moore bound (as shown in the table below). <br />
<br />
Jana Šiagiová and Jozef Širáň have found a general construction for Cayley graphs. SS graphs are constructed using semi-direct products of a product of finite fields and ''Z<sub>2</sub>'', where each such group yields a range of graphs of different degrees. SS graphs are the largest known Cayley graphs for degree larger than 30 and diameter 2, with order up to about 50% of the Moore bound.<br />
<br />
Marcel Abas has found a general construction for Cayley graphs of any degree with order half of the Moore bound using direct product of dihedral groups <i>D<sub>m</sub></i> with cyclic groups <i>Z<sub>n</sub></i>. It has been shown that, in asymptotic sense, the most of record Cayley graphs of diameter two is obtained by Abas construction. Using semidirect product of <i>Z<sub>n</sub></i>&times;<i>Z<sub>n</sub></i> with <i>Z<sub>2</sub></i> he has found (for degrees <i>13 &le; d &le; 57</i>) largest known Cayley graphs in 34 cases of total 45 degrees and he constructed Cayley graphs of diameter two and of order of 0.684 of the Moore bound for every degree <i>d &ge; 360 756</i>.<br />
<br />
A range of Cayley graphs of diameter 2 and degree larger than 12 was found by Eyal Loz using semi-direct products of cyclic groups.<br />
<br />
<center><br />
{| border="0" cellspacing="0" cellpadding="0"<br />
|-<br />
|<br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
! '''<math>d</math>''' !! '''Order''' !! '''% Moore Bound'''<br />
|-<br />
| '''4''' ||style="background-color: #ccff33; text-align: center;" | '''13''' || 76.47%<br />
|-<br />
| '''5''' ||style="background-color: #ccff33; text-align: center;" | '''18''' || 69.23%<br />
|-<br />
| '''6''' ||style="background-color: #ccff33; text-align: center;" | '''32''' || 86.48%<br />
|-<br />
| '''7''' ||style="background-color: #ccff33; text-align: center;" | '''36''' || 72% <br />
|-<br />
| '''8''' ||style="background-color: #ccff33; text-align: center;" | '''48''' || 73.84%<br />
|-<br />
| '''9''' ||style="background-color: #ccff33; text-align: center;" | '''60''' || 73.17%<br />
|-<br />
| '''10''' ||style="background-color: #ccff33; text-align: center;" | '''72''' || 71.28%<br />
|-<br />
| '''11''' ||style="background-color: #ccff33; text-align: center;" | '''84''' || 68.85%<br />
|-<br />
| '''12''' ||style="background-color: #ccff33; text-align: center;" | '''96''' || 66.20%<br />
|-<br />
| '''13''' ||style="background-color: #FF9900; text-align: center;" | 112 || 65.88%<br />
|-<br />
| '''14''' ||style="background-color: #FF9900; text-align: center;" | 128 || 64.97%<br />
|-<br />
| '''15''' ||style="background-color: #FF9900; text-align: center;" | 144 || 63.71%<br />
|-<br />
| '''16''' ||style="background-color: #FFFF00; text-align: center;" | 200 || 77.82%<br />
|-<br />
| '''17''' ||style="background-color: #FFFF00; text-align: center;" | 200 || 68.96% <br />
|-<br />
| '''18''' ||style="background-color: #FFFF00; text-align: center;" | 200 || 61.53% <br />
|-<br />
| '''19''' ||style="background-color: #FF9900; text-align: center;" | 200 || 55.24%<br />
|-<br />
| '''20''' ||style="background-color: #FF9900; text-align: center;" | 210 || 52.36%<br />
|-<br />
| '''21''' ||style="background-color: #FFFF00; text-align: center;" | 288 || 65.15%<br />
|-<br />
| '''22''' ||style="background-color: #FFFF00; text-align: center;" | 288 || 59.38%<br />
|-<br />
| '''23''' ||style="background-color: #FFFF00; text-align: center;" | 392 || 73.96%<br />
|-<br />
| '''24''' ||style="background-color: #FFFF00; text-align: center;" | 392 || 67.93%<br />
|-<br />
| '''25''' ||style="background-color: #FFFF00; text-align: center;" | 392 || 62.61%<br />
|-<br />
| '''26''' ||style="background-color: #FFFF00; text-align: center;" | 392 || 57.90%<br />
|-<br />
| '''27''' ||style="background-color: #FFFF00; text-align: center;" | 392 || 53.69%<br />
|-<br />
| '''28''' ||style="background-color: #FFFF00; text-align: center;" | 512 || 65.22%<br />
|-<br />
| '''29''' ||style="background-color: #FFFF00; text-align: center;" | 512 || 60.80% <br />
|-<br />
| '''30''' ||style="background-color: #FFFF00; text-align: center;" | 512 || 56.82%<br />
|-<br />
|}<br />
<br />
|| <br />
<br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
! '''<math>d</math>''' !! '''Order''' !! '''% Moore Bound'''<br />
|-<br />
| '''31''' ||style="background-color: #FFFF00; text-align: center;" | 648 || 67.35%<br />
|-<br />
| '''32''' ||style="background-color: #FFFF00; text-align: center;" | 648 || 63.21%<br />
|-<br />
| '''33''' ||style="background-color: #FFFF00; text-align: center;" | 648 || 59.44%<br />
|-<br />
| '''34''' ||style="background-color: #FFFF00; text-align: center;" | 648 || 56.00% <br />
|-<br />
| '''35''' ||style="background-color: #FFFF00; text-align: center;" | 648 || 52.85%<br />
|-<br />
| '''36''' ||style="background-color: #00F3F7; text-align: center;" | 648 || 49.96%<br />
|-<br />
| '''37''' ||style="background-color: #FFFF00; text-align: center;" | 800 || 58.39%<br />
|-<br />
| '''38''' ||style="background-color: #FFFF00; text-align: center;" | 800 || 55.36%<br />
|-<br />
| '''39''' ||style="background-color: #FFFF00; text-align: center;" | 800 || 52.56%<br />
|-<br />
| '''40''' ||style="background-color: #FFFF00; text-align: center;" | 968 || 60.46% <br />
|-<br />
| '''41''' ||style="background-color: #FFFF00; text-align: center;" | 968 || 57.55%<br />
|-<br />
| '''42''' ||style="background-color: #FFFF00; text-align: center;" | 968 || 54.84% <br />
|-<br />
| '''43''' ||style="background-color: #FFFF00; text-align: center;" | 968 || 52.32%<br />
|-<br />
| '''44''' ||style="background-color: #00F3F7; text-align: center;" | 968 || 49.97%<br />
|-<br />
| '''45''' ||style="background-color: #99cc33; text-align: center;" | 1058 || 52.22%<br />
|-<br />
| '''46''' ||style="background-color: #FFFF00; text-align: center;" | 1152 || 54.41% <br />
|-<br />
| '''47''' ||style="background-color: #FFFF00; text-align: center;" | 1152 || 52.12%<br />
|-<br />
| '''48''' ||style="background-color: #FFFF00; text-align: center;" | 1152 || 49.97%<br />
|-<br />
| '''49''' ||style="background-color: #FFFF00; text-align: center;" | 1352 || 56.28%<br />
|-<br />
| '''50''' ||style="background-color: #FFFF00; text-align: center;" | 1352 || 54.05%<br />
|-<br />
| '''51''' ||style="background-color: #FFFF00; text-align: center;" | 1352 || 51.96% <br />
|-<br />
| '''52''' ||style="background-color: #00F3F7; text-align: center;" | 1352 || 49.98%<br />
|-<br />
| '''53''' ||style="background-color: #99cc33; text-align: center;" | 1458 || 51.88%<br />
|-<br />
| '''54''' ||style="background-color: #FFFF00; text-align: center;" | 1568 || 53.75% <br />
|-<br />
| '''55''' ||style="background-color: #FFFF00; text-align: center;" | 1568 || 51.81% <br />
|-<br />
| '''56''' ||style="background-color: #FFFF00; text-align: center;" | 1568 || 49.98%<br />
|-<br />
| '''57''' ||style="background-color: #99cc33; text-align: center;" | 1682 || 51.75%<br />
|-<br />
|}<br />
|-<br />
|}<br />
<br />
<br />
<br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: #ccff33; text-align: center;" | * || Optimal Cayley Graphs found by Marston Conder.<br />
|-<br />
|style="background-color: #FF9900; text-align: center;" | * || Cayley Graphs found by Eyal Loz.<br />
|-<br />
|style="background-color: #99cc33; text-align: center;" | * || Cayley Graphs found by Jana Šiagiová and Jozef Širáň.<br />
|-<br />
|style="background-color: #00F3F7; text-align: center;" | * || Cayley Graphs found by Marcel Abas.<br />
|-<br />
|style="background-color: #FFFF00; text-align: center;" | * || Cayley Graphs found by Marcel Abas.<br />
|-<br />
|}<br />
</center><br />
<br />
==References==<br />
* Marcel Abas, "Cayley graphs of diameter two and any degree with order half of the Moore bound", Discrete Applied Mathematics, Volume 173, 20 August 2014, Pages 1-7<br />
* Marcel Abas, "Cayley graphs of diameter two with order greater than 0.684 of the Moore bound for any degree", European Journal of Combinatorics, Volume 57, (2016), Pages 109-120, [https://arxiv.org/pdf/1511.03706.pdf, PDF version]<br />
* Marcel Abas, "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]<br />
* J. Dinneen, Michael; Hafner, Paul 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] <br />
* Miller, Mirka; Širáň, Jozef (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] <br />
* Loz, Eyal; Širáň, Jozef (2008), "New record graphs in the degree-diameter problem", Australasian Journal of Combinatorics 41: 63–80<br />
* 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.<br />
* Eugene Curtin, Cubic Cayley graphs with small diameter Discrete Mathematics and Theoretical Computer Science 4, 2001, 123-132, [http://www.emis.de/journals/DMTCS/volumes/abstracts/pdfpapers/dm040205.pdf PDF version]<br />
* Jana Šiagiová and Jozef Širáň, "A note on large Cayley graphs of diameter two and given degree", Discrete Mathematics, Volume 305, Issues 1-3, 6 December 2005, Pages 379-382<br />
* Randall Dougherty and Vance Faber, "The Degree-Diameter Problem for Several Varieties of Cayley Graphs I: The Abelian Case", SIAM Journal on Discrete Mathematics, Volume 17 , Issue 3, 2004, 478-519, ISSN:0895-4801<br />
* R. Stanton and D. Cowan, "Note on a “square” functional equation", SIAM Rev., 12 (1970), pp. 277–279.<br />
<br />
==External links==<br />
* [http://www-mat.upc.es/grup_de_grafs/ Degree Diameter] online table.<br />
* [http://people.csse.uwa.edu.au/gordon/data.html Gordon Royle's] data pages.<br />
* [http://www.eyal.com.au/wiki/The_Degree/Diameter_Problem Eyal Loz's] Degree-Diameter problem page.<br />
<br />
<br />
<br />
[[Category:The Degree/Diameter Problem]]</div>
Guillermo
http://combinatoricswiki.org/index.php?title=The_Degree_Diameter_Problem_for_General_Graphs&diff=646
The Degree Diameter Problem for General Graphs
2022-02-18T05:43:37Z
<p>Guillermo: /* Table of the orders of the largest known graphs for the undirected degree diameter problem */</p>
<hr />
<div>==Introduction==<br />
The '''degree/diameter problem for general graphs''' can be stated as follows:<br />
<br />
''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''.''<br />
<br />
In attempting to settle the values of ''N(d,k)'', research activities in this problem have follow the following two directions:<br />
<br />
*Increasing the lower bounds for ''N(d,k)'' by constructing ever larger graphs.<br />
<br />
* Lowering and/or setting upper bounds for ''N(d,k)'' by proving the non-existence of graphs<br />
whose order is close to the Moore bounds ''M(d,k)=(d(d-1)<sup>k</sup>-2)(d-2)<sup>-1</sup>''.<br />
<br />
==Increasing the lower bounds for ''N(d,k)''==<br />
<br />
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.<br />
<br />
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&nbsp;≤&nbsp;''d''&nbsp;≤&nbsp;20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;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.<br />
<br />
<br />
===Table of the orders of the largest known graphs for the undirected degree diameter problem===<br />
<br />
<center> <br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
| '''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 <br />
|-<br />
<br />
| '''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 <br />
|-<br />
| '''5''' ||style="background-color: blue;" | '''24''' ||style="background-color: #ffff00;" | 72 ||style="background-color: #ffff00;" | 212 ||style="background-color: #FF9900;" | 624 ||style="background-color: #7BB661;" | 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 <br />
|-<br />
| '''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: #7BB661;" | 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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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: #7BB661;" | 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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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<br />
|-<br />
| '''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: #7BB661;" | 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<br />
|-<br />
| '''13''' ||style="background-color: #ff99ff;" | 162 ||style="background-color: #FFF8DC;" | 856 ||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<br />
|-<br />
| '''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: #7BB661;" | 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<br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|}<br />
</center><br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="3" cellpadding="3" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: blue; text-align: center;" | * || Other non Moore but optimal graphs. <br />
|-<br />
|style="background-color: #999900; text-align: center;" | * || Graph found by J. Allwright.<br />
|-<br />
|style="background-color: #336666; text-align: center;" | * || Graph found by G. Wegner.<br />
|-<br />
|style="background-color: #ffff00; text-align: center;" | * || Graphs found by G. Exoo.<br />
|-<br />
|style="background-color: #FFF8DC; text-align: center;" | * || Graphs found by V. Pelekhaty. The adjacency list can be found here [[File:Pelekhaty-856-13-3.pdf]].<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #666633; text-align: center;" | * || Graphs found by J. Gómez. <br />
|-<br />
|style="background-color: #993300; text-align: center;" | * || Graph found by M. Mitjana and F. Comellas. This graph was also found independently by M. Sampels.<br />
|-<br />
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by C. Delorme.<br />
|-<br />
|style="background-color: #6666ff; text-align: center;" | * || Graphs found by C. Delorme and G. Farhi.<br />
|-<br />
|style="background-color: #187eac; text-align: center;" | * || Graphs found by E. Canale. (2012)<br />
|-<br />
|style="background-color: #3399cc; text-align: center;" | * || Graph found by J. C. Bermond, C. Delorme, and G. Farhi<br />
|-<br />
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by J. Gómez and M. A. Fiol.<br />
|-<br />
|style="background-color: #999999; text-align: center;" | * || Graphs found by J. Gómez, M. A. Fiol, and O. Serra.<br />
|-<br />
|style="background-color: #66cc66; text-align: center;" | * || Graph found by M.A. Fiol and J.L.A. Yebra.<br />
|-<br />
|style="background-color: #666666; text-align: center;" | * || Graph found by F. Comellas and J. Gómez.<br />
|-<br />
|style="background-color: #ddff00; text-align: center;" | * || Graph found by Jianxiang Chen.<br />
|-<br />
|style="background-color: #7BB661; 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.<br />
|-<br />
|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. <br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #aa8268; text-align: center;" | * || Graphs found by A. Rodriguez. (2012)<br />
|-<br />
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by M. Sampels.<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #ffcc99; text-align: center;" | * || Graph found by M. Conder.<br />
|-<br />
|style="background-color: #cc0033; text-align: center;" | * || Graphs found by Brown, W. G. (1966).<br />
|-<br />
|style="background-color: #99FF00; text-align: center;" | * || Graph found by M. Abas. (2017). More details are available in a paper by the author.<br />
|}<br />
</center><br />
<br />
==Lowering and/or setting upper bounds for ''N(d,k)''== <br />
<br />
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)-&delta;'' for small ''&delta;''. The parameter ''&delta;'' is called the defect. Such graphs are called ''(d,k,-&delta;)''-graphs.<br />
<br />
For ''&delta;=1'' the only ''(d,k,-1)''-graphs are the cycles on ''2k'' vertices. Erd&ouml;s, Fajtlowitcz and Hoffman, who proved the non-existence of ''(d,2,-1)''-graphs for ''d&ne;3''. Then, Bannai and Ito, and also<br />
independently, Kurosawa and Tsujii, proved the non-existence of ''(d,k,-1)''-graphs for ''d&ge;3'' and ''k&ge;3''.<br />
<br />
For ''&delta;=2'', the ''(2,k,-2)''-graphs are the cycles on ''2k-1''. Considering ''d&ge;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. <br />
<br />
When ''&delta;=2'', ''d&ge;3'' and ''k&ge;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&ge;4'' by Leif Jorgensen, the non-existence of ''(4,k,-2)''-graphs with ''k&ge;3'' by Mirka Miller and Rino Simanjuntak, some structural properties of ''(5,k,-2)''-graphs with ''k&ge;3'' by Guillermo Pineda-Villavicencio and Mirka Miller, the obtaining of several necessary conditions for the existence of ''(d,2,-2)''-graphs with ''d&ge;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.<br />
<br />
For the case of ''&delta;&ge;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&ge;2'' by Guillermo Pineda-Villavicencio and Mirka Miller by proving the non-existence of ''(3,k,-4)''-graphs with ''k&ge;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. <br />
<br />
===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===<br />
<br />
<center><br />
{| border="1"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
|'''3'''<br />
| align="center" |<br />
{| border="2" style="background:red;" <br />
|10<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|20<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#336666;" <br />
|38<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|92<br />
|-<br />
|76.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|188<br />
|-<br />
|70.21%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|380<br />
|-<br />
|51.57%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|764<br />
|-<br />
|43.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|1532<br />
|-<br />
|39.16%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|3068<br />
|-<br />
|40.74%<br />
|}<br />
|-<br />
|'''4'''<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|15<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52<br />
|-<br />
|78.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160<br />
|-<br />
|61.25%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484<br />
|-<br />
|75.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1456<br />
|-<br />
|50.82%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4372<br />
|-<br />
|30.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13120<br />
|-<br />
|24.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|39364<br />
|-<br />
|19.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|118096<br />
|-<br />
|14.99%<br />
|}<br />
|-<br />
|'''5'''<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|24<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|104<br />
|-<br />
|69.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|424<br />
|-<br />
|50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1704<br />
|-<br />
|36.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6824<br />
|-<br />
|40.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27304<br />
|-<br />
|20.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|109224<br />
|-<br />
|15.59%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|436904<br />
|-<br />
|13.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1747624<br />
|-<br />
|10.70%<br />
|}<br />
|-<br />
|'''6'''<br />
| align="center" |<br />
{| border="2" style="background:#336666;" <br />
|32<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|186<br />
|-<br />
|59.67%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|936<br />
|-<br />
|41.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4686<br />
|-<br />
|29.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23436<br />
|-<br />
|33.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117186<br />
|-<br />
|16.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|585936<br />
|-<br />
|13.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929686<br />
|-<br />
|10.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14648436<br />
|-<br />
|8.55%<br />
|}<br />
|-<br />
|'''7'''<br />
| align="center" |<br />
{| border="2" style="background:red;" <br />
|50<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|301<br />
|-<br />
|55.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1813<br />
|-<br />
|37.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|10885<br />
|-<br />
|25.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|65317<br />
|-<br />
|18.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|391909<br />
|-<br />
|13.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2351461<br />
|-<br />
|10.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14108773<br />
|-<br />
|8.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|84652645<br />
|-<br />
|7.09%<br />
|}<br />
|-<br />
|'''8'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|63<br />
|-<br />
|90.47%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|456<br />
|-<br />
|55.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3200<br />
|-<br />
|34.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|22408<br />
|-<br />
|22.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|156864<br />
|-<br />
|25.29%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1098056<br />
|-<br />
|11.94%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7686400<br />
|-<br />
|9.56%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53804808<br />
|-<br />
|7.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|376633664<br />
|-<br />
|6.61%<br />
|}<br />
|-<br />
|'''9'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|80<br />
|-<br />
|92.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|657<br />
|-<br />
|89.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5265<br />
|-<br />
|29.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42129<br />
|-<br />
|19.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|337041<br />
|-<br />
|22.51%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2696337<br />
|-<br />
|10.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|21570705<br />
|-<br />
|7.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|172565649<br />
|-<br />
|7.02%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1380525201<br />
|-<br />
|4.77%<br />
|}<br />
|-<br />
|'''10'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|99<br />
|-<br />
|91.91%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|910<br />
|-<br />
|71.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8200<br />
|-<br />
|27.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73810<br />
|-<br />
|17.80%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|664300<br />
|-<br />
|20.27%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5978710<br />
|-<br />
|9.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53808400<br />
|-<br />
|7.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484275610<br />
|-<br />
|5.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4358480500<br />
|-<br />
|4.61%<br />
|}<br />
|-<br />
|'''11'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|120<br />
|-<br />
|86.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1221<br />
|-<br />
|58.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12221<br />
|-<br />
|26.18%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122221<br />
|-<br />
|15.95%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222221<br />
|-<br />
|12.83%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222221<br />
|-<br />
|8.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122222221<br />
|-<br />
|6.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222222221<br />
|-<br />
|5.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222222221<br />
|-<br />
|4.91%<br />
|}<br />
|-<br />
|'''12'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|143<br />
|-<br />
|93%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1596<br />
|-<br />
|49.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|17568<br />
|-<br />
|26.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|193260<br />
|-<br />
|15.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2125872<br />
|-<br />
|16.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23384604<br />
|-<br />
|8.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|257230656<br />
|-<br />
|6.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2829537228<br />
|-<br />
|5.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|31124909520<br />
|-<br />
|4.83%<br />
|}<br />
|-<br />
|'''13'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|168<br />
|-<br />
|96.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2041<br />
|-<br />
|41.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|24505<br />
|-<br />
|26.77%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|294073<br />
|-<br />
|13.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3528889<br />
|-<br />
|15.05%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42346681<br />
|-<br />
|7.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|508160185<br />
|-<br />
|5.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6097922233<br />
|-<br />
|4.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73175066809<br />
|-<br />
|4.20%<br />
|}<br />
|-<br />
|'''14'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195<br />
|-<br />
|93.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2562<br />
|-<br />
|35.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|33320<br />
|-<br />
|24.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|433174<br />
|-<br />
|13.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5631276<br />
|-<br />
|14.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73206602<br />
|-<br />
|8.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|951685840<br />
|-<br />
|5.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12371915934<br />
|-<br />
|4.85%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160834907156<br />
|-<br />
|4.37%<br />
|}<br />
|-<br />
|'''15'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|224<br />
|-<br />
|83.03%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3165<br />
|-<br />
|38.38%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|44325<br />
|-<br />
|26.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|620565<br />
|-<br />
|12.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8687925<br />
|-<br />
|16.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|121630965<br />
|-<br />
|7.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1702833525<br />
|-<br />
|5.28%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23839669365<br />
|-<br />
|4.91%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|333755371125<br />
|-<br />
|2.99%<br />
|}<br />
|-<br />
|'''16'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|255<br />
|-<br />
|77.64%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3856<br />
|-<br />
|41.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|57856<br />
|-<br />
|25.30%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|867856<br />
|-<br />
|15.26%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13017856<br />
|-<br />
|13.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195267856<br />
|-<br />
|7.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929017856<br />
|-<br />
|4.79%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|43935267856<br />
|-<br />
|4.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|659029017856<br />
|-<br />
|1.96%<br />
|}<br />
|-<br />
|'''17'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|288<br />
|-<br />
|95.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4641<br />
|-<br />
|34.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|74273<br />
|-<br />
|25.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1188385<br />
|-<br />
|11.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|19014177<br />
|-<br />
|16.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|304226849<br />
|-<br />
|6.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4867629601<br />
|-<br />
|4.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|77882073633<br />
|-<br />
|4.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1246113178145<br />
|-<br />
|1.22%<br />
|}<br />
|-<br />
|'''18'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|323<br />
|-<br />
|95.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5526<br />
|-<br />
|29.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|93960<br />
|-<br />
|25.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1597338<br />
|-<br />
|10.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27154764<br />
|-<br />
|14.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|461631006<br />
|-<br />
|5.74%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7847727120<br />
|-<br />
|4.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|133411361058<br />
|-<br />
|4.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2267993138004<br />
|-<br />
|0.73%<br />
|}<br />
|-<br />
|'''19'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|360<br />
|-<br />
|93.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6517<br />
|-<br />
|25.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117325<br />
|-<br />
|20.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2111869<br />
|-<br />
|10.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|38013661<br />
|-<br />
|10.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|684245917<br />
|-<br />
|5.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12316426525<br />
|-<br />
|4.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|221695677469<br />
|-<br />
|3.99%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3990522194461<br />
|-<br />
|0.45%<br />
|}<br />
|-<br />
|'''20'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|399<br />
|-<br />
|95.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7620<br />
|-<br />
|25.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|144800<br />
|-<br />
|24.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2751220<br />
|-<br />
|10.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52273200<br />
|-<br />
|17.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|993190820<br />
|-<br />
|5.6%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|18870625600<br />
|-<br />
|4.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|358541886420<br />
|-<br />
|3.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6812295842000<br />
|-<br />
|1.14%<br />
|}<br />
|}<br />
</center><br />
<br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: red; text-align: center;" | * || The Moore bound.<br />
|-<br />
|style="background-color: #ABCDEF; text-align: center;" | * || Upper bound introduced by A. Hoffman, R. Singleton, Bannai, E. and Ito, T. <br />
|-<br />
|style="background-color: #999900; text-align: center;" | * || Upper bound introduced by Leif Jorgensen.<br />
|-<br />
|style="background-color: #336666; text-align: center;" | * || Optimal graphs found by Buset and by Molodtsov. <br />
|-<br />
|style="background-color: #ffff00; text-align: center;" | * || Graphs shown optimal.<br />
|-<br />
|}<br />
</center><br />
<br />
==References==<br />
* 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]<br />
* Bannai, E.; Ito, T. (1981), "Regular graphs with excess one", Discrete Mathematics 37:147-158, doi:10.1016/0012-365X(81)90215-6.<br />
* 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.<br />
* 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].<br />
* 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.<br />
* Erd&ouml;s P; Fajtlowicz, S.; Hoffman A. J. (1980), "Maximum degree in graphs of diameter 2", Networks 10: 87-90.<br />
* 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]. <br />
* L. K. Jorgensen (1992), "Diameters of cubic graphs", Discrete Applied Mathematics 37/38: 347-351, doi:10.1016/0166-218X(92)90144-Y.<br />
* 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.<br />
* Kurosawa, K.; Tsujii, S. (1981), "Considerations on diameter of communication networks", Electronics and Communications in Japan 64A (4): 37-45.<br />
* Loz, E.; Širáň, J. (2008), "New record graphs in the degree-diameter problem", Australasian Journal of Combinatorics 41: 63–80.<br />
* 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.<br />
* 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.<br />
* 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.<br />
* 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.<br />
* 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].<br />
* Molodtsov, S. G. (2006), "Largest Graphs of Diameter 2 and Maximum Degree 6", Lecture Notes in Computer Science 4123: 853-857.<br />
* 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.<br />
* Pineda-Villavicencio, G.; Miller, M., "Complete characterization of graphs of maximum degree 3 and defect at most 4", submitted.<br />
* 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. <br />
* 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.<br />
* Brown, W. G. (1966) On graphs that do not contain a Thomsen graph. Canadian Mathematical Bulletin, 9, 281 - 285.<br />
<br />
==External links==<br />
* [http://www-mat.upc.es/grup_de_grafs/ Degree Diameter] online table.<br />
* [http://www.eyal.com.au/wiki/The_Degree/Diameter_Problem Eyal Loz's] Degree-Diameter problem page.<br />
* [http://isu.indstate.edu/ge/DD/index.html Geoffrey Exoo's] Degree-Diameter record graphs page.<br />
<br />
<br />
[[Category:The Degree/Diameter Problem]]</div>
Guillermo
http://combinatoricswiki.org/index.php?title=The_Degree_Diameter_Problem_for_General_Graphs&diff=645
The Degree Diameter Problem for General Graphs
2022-02-18T05:41:42Z
<p>Guillermo: /* Table of the orders of the largest known graphs for the undirected degree diameter problem */</p>
<hr />
<div>==Introduction==<br />
The '''degree/diameter problem for general graphs''' can be stated as follows:<br />
<br />
''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''.''<br />
<br />
In attempting to settle the values of ''N(d,k)'', research activities in this problem have follow the following two directions:<br />
<br />
*Increasing the lower bounds for ''N(d,k)'' by constructing ever larger graphs.<br />
<br />
* Lowering and/or setting upper bounds for ''N(d,k)'' by proving the non-existence of graphs<br />
whose order is close to the Moore bounds ''M(d,k)=(d(d-1)<sup>k</sup>-2)(d-2)<sup>-1</sup>''.<br />
<br />
==Increasing the lower bounds for ''N(d,k)''==<br />
<br />
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.<br />
<br />
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&nbsp;≤&nbsp;''d''&nbsp;≤&nbsp;20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;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.<br />
<br />
<br />
===Table of the orders of the largest known graphs for the undirected degree diameter problem===<br />
<br />
<center> <br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
| '''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 <br />
|-<br />
<br />
| '''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 <br />
|-<br />
| '''5''' ||style="background-color: blue;" | '''24''' ||style="background-color: #ffff00;" | 72 ||style="background-color: #ffff00;" | 212 ||style="background-color: #FF9900;" | 624 ||style="background-color: #7BB661;" | 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 <br />
|-<br />
| '''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: #7BB661;" | 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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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: #7BB661;" | 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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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<br />
|-<br />
| '''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: #7BB661;" | 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<br />
|-<br />
| '''13''' ||style="background-color: #ff99ff;" | 162 ||style="background-color: #FFF8DC;" | 856 ||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<br />
|-<br />
| '''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: #7BB661;" | 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<br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|}<br />
</center><br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="3" cellpadding="3" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: blue; text-align: center;" | * || Other non Moore but optimal graphs. <br />
|-<br />
|style="background-color: #999900; text-align: center;" | * || Graph found by J. Allwright.<br />
|-<br />
|style="background-color: #336666; text-align: center;" | * || Graph found by G. Wegner.<br />
|-<br />
|style="background-color: #ffff00; text-align: center;" | * || Graphs found by G. Exoo.<br />
|-<br />
|style="background-color: #FFF8DC; text-align: center;" | * || Graphs found by V. Pelekhaty. The adjacency list can be found here [[Pelekhaty-856-13-3.pdf]]<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #666633; text-align: center;" | * || Graphs found by J. Gómez. <br />
|-<br />
|style="background-color: #993300; text-align: center;" | * || Graph found by M. Mitjana and F. Comellas. This graph was also found independently by M. Sampels.<br />
|-<br />
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by C. Delorme.<br />
|-<br />
|style="background-color: #6666ff; text-align: center;" | * || Graphs found by C. Delorme and G. Farhi.<br />
|-<br />
|style="background-color: #187eac; text-align: center;" | * || Graphs found by E. Canale. (2012)<br />
|-<br />
|style="background-color: #3399cc; text-align: center;" | * || Graph found by J. C. Bermond, C. Delorme, and G. Farhi<br />
|-<br />
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by J. Gómez and M. A. Fiol.<br />
|-<br />
|style="background-color: #999999; text-align: center;" | * || Graphs found by J. Gómez, M. A. Fiol, and O. Serra.<br />
|-<br />
|style="background-color: #66cc66; text-align: center;" | * || Graph found by M.A. Fiol and J.L.A. Yebra.<br />
|-<br />
|style="background-color: #666666; text-align: center;" | * || Graph found by F. Comellas and J. Gómez.<br />
|-<br />
|style="background-color: #ddff00; text-align: center;" | * || Graph found by Jianxiang Chen.<br />
|-<br />
|style="background-color: #7BB661; 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.<br />
|-<br />
|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. <br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #aa8268; text-align: center;" | * || Graphs found by A. Rodriguez. (2012)<br />
|-<br />
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by M. Sampels.<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #ffcc99; text-align: center;" | * || Graph found by M. Conder.<br />
|-<br />
|style="background-color: #cc0033; text-align: center;" | * || Graphs found by Brown, W. G. (1966).<br />
|-<br />
|style="background-color: #99FF00; text-align: center;" | * || Graph found by M. Abas. (2017). More details are available in a paper by the author.<br />
|}<br />
</center><br />
<br />
==Lowering and/or setting upper bounds for ''N(d,k)''== <br />
<br />
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)-&delta;'' for small ''&delta;''. The parameter ''&delta;'' is called the defect. Such graphs are called ''(d,k,-&delta;)''-graphs.<br />
<br />
For ''&delta;=1'' the only ''(d,k,-1)''-graphs are the cycles on ''2k'' vertices. Erd&ouml;s, Fajtlowitcz and Hoffman, who proved the non-existence of ''(d,2,-1)''-graphs for ''d&ne;3''. Then, Bannai and Ito, and also<br />
independently, Kurosawa and Tsujii, proved the non-existence of ''(d,k,-1)''-graphs for ''d&ge;3'' and ''k&ge;3''.<br />
<br />
For ''&delta;=2'', the ''(2,k,-2)''-graphs are the cycles on ''2k-1''. Considering ''d&ge;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. <br />
<br />
When ''&delta;=2'', ''d&ge;3'' and ''k&ge;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&ge;4'' by Leif Jorgensen, the non-existence of ''(4,k,-2)''-graphs with ''k&ge;3'' by Mirka Miller and Rino Simanjuntak, some structural properties of ''(5,k,-2)''-graphs with ''k&ge;3'' by Guillermo Pineda-Villavicencio and Mirka Miller, the obtaining of several necessary conditions for the existence of ''(d,2,-2)''-graphs with ''d&ge;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.<br />
<br />
For the case of ''&delta;&ge;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&ge;2'' by Guillermo Pineda-Villavicencio and Mirka Miller by proving the non-existence of ''(3,k,-4)''-graphs with ''k&ge;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. <br />
<br />
===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===<br />
<br />
<center><br />
{| border="1"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
|'''3'''<br />
| align="center" |<br />
{| border="2" style="background:red;" <br />
|10<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|20<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#336666;" <br />
|38<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|92<br />
|-<br />
|76.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|188<br />
|-<br />
|70.21%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|380<br />
|-<br />
|51.57%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|764<br />
|-<br />
|43.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|1532<br />
|-<br />
|39.16%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|3068<br />
|-<br />
|40.74%<br />
|}<br />
|-<br />
|'''4'''<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|15<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52<br />
|-<br />
|78.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160<br />
|-<br />
|61.25%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484<br />
|-<br />
|75.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1456<br />
|-<br />
|50.82%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4372<br />
|-<br />
|30.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13120<br />
|-<br />
|24.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|39364<br />
|-<br />
|19.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|118096<br />
|-<br />
|14.99%<br />
|}<br />
|-<br />
|'''5'''<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|24<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|104<br />
|-<br />
|69.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|424<br />
|-<br />
|50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1704<br />
|-<br />
|36.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6824<br />
|-<br />
|40.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27304<br />
|-<br />
|20.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|109224<br />
|-<br />
|15.59%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|436904<br />
|-<br />
|13.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1747624<br />
|-<br />
|10.70%<br />
|}<br />
|-<br />
|'''6'''<br />
| align="center" |<br />
{| border="2" style="background:#336666;" <br />
|32<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|186<br />
|-<br />
|59.67%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|936<br />
|-<br />
|41.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4686<br />
|-<br />
|29.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23436<br />
|-<br />
|33.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117186<br />
|-<br />
|16.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|585936<br />
|-<br />
|13.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929686<br />
|-<br />
|10.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14648436<br />
|-<br />
|8.55%<br />
|}<br />
|-<br />
|'''7'''<br />
| align="center" |<br />
{| border="2" style="background:red;" <br />
|50<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|301<br />
|-<br />
|55.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1813<br />
|-<br />
|37.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|10885<br />
|-<br />
|25.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|65317<br />
|-<br />
|18.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|391909<br />
|-<br />
|13.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2351461<br />
|-<br />
|10.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14108773<br />
|-<br />
|8.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|84652645<br />
|-<br />
|7.09%<br />
|}<br />
|-<br />
|'''8'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|63<br />
|-<br />
|90.47%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|456<br />
|-<br />
|55.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3200<br />
|-<br />
|34.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|22408<br />
|-<br />
|22.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|156864<br />
|-<br />
|25.29%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1098056<br />
|-<br />
|11.94%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7686400<br />
|-<br />
|9.56%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53804808<br />
|-<br />
|7.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|376633664<br />
|-<br />
|6.61%<br />
|}<br />
|-<br />
|'''9'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|80<br />
|-<br />
|92.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|657<br />
|-<br />
|89.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5265<br />
|-<br />
|29.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42129<br />
|-<br />
|19.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|337041<br />
|-<br />
|22.51%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2696337<br />
|-<br />
|10.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|21570705<br />
|-<br />
|7.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|172565649<br />
|-<br />
|7.02%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1380525201<br />
|-<br />
|4.77%<br />
|}<br />
|-<br />
|'''10'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|99<br />
|-<br />
|91.91%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|910<br />
|-<br />
|71.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8200<br />
|-<br />
|27.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73810<br />
|-<br />
|17.80%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|664300<br />
|-<br />
|20.27%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5978710<br />
|-<br />
|9.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53808400<br />
|-<br />
|7.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484275610<br />
|-<br />
|5.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4358480500<br />
|-<br />
|4.61%<br />
|}<br />
|-<br />
|'''11'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|120<br />
|-<br />
|86.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1221<br />
|-<br />
|58.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12221<br />
|-<br />
|26.18%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122221<br />
|-<br />
|15.95%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222221<br />
|-<br />
|12.83%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222221<br />
|-<br />
|8.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122222221<br />
|-<br />
|6.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222222221<br />
|-<br />
|5.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222222221<br />
|-<br />
|4.91%<br />
|}<br />
|-<br />
|'''12'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|143<br />
|-<br />
|93%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1596<br />
|-<br />
|49.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|17568<br />
|-<br />
|26.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|193260<br />
|-<br />
|15.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2125872<br />
|-<br />
|16.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23384604<br />
|-<br />
|8.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|257230656<br />
|-<br />
|6.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2829537228<br />
|-<br />
|5.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|31124909520<br />
|-<br />
|4.83%<br />
|}<br />
|-<br />
|'''13'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|168<br />
|-<br />
|96.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2041<br />
|-<br />
|41.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|24505<br />
|-<br />
|26.77%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|294073<br />
|-<br />
|13.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3528889<br />
|-<br />
|15.05%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42346681<br />
|-<br />
|7.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|508160185<br />
|-<br />
|5.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6097922233<br />
|-<br />
|4.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73175066809<br />
|-<br />
|4.20%<br />
|}<br />
|-<br />
|'''14'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195<br />
|-<br />
|93.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2562<br />
|-<br />
|35.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|33320<br />
|-<br />
|24.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|433174<br />
|-<br />
|13.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5631276<br />
|-<br />
|14.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73206602<br />
|-<br />
|8.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|951685840<br />
|-<br />
|5.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12371915934<br />
|-<br />
|4.85%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160834907156<br />
|-<br />
|4.37%<br />
|}<br />
|-<br />
|'''15'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|224<br />
|-<br />
|83.03%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3165<br />
|-<br />
|38.38%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|44325<br />
|-<br />
|26.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|620565<br />
|-<br />
|12.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8687925<br />
|-<br />
|16.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|121630965<br />
|-<br />
|7.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1702833525<br />
|-<br />
|5.28%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23839669365<br />
|-<br />
|4.91%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|333755371125<br />
|-<br />
|2.99%<br />
|}<br />
|-<br />
|'''16'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|255<br />
|-<br />
|77.64%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3856<br />
|-<br />
|41.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|57856<br />
|-<br />
|25.30%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|867856<br />
|-<br />
|15.26%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13017856<br />
|-<br />
|13.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195267856<br />
|-<br />
|7.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929017856<br />
|-<br />
|4.79%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|43935267856<br />
|-<br />
|4.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|659029017856<br />
|-<br />
|1.96%<br />
|}<br />
|-<br />
|'''17'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|288<br />
|-<br />
|95.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4641<br />
|-<br />
|34.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|74273<br />
|-<br />
|25.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1188385<br />
|-<br />
|11.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|19014177<br />
|-<br />
|16.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|304226849<br />
|-<br />
|6.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4867629601<br />
|-<br />
|4.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|77882073633<br />
|-<br />
|4.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1246113178145<br />
|-<br />
|1.22%<br />
|}<br />
|-<br />
|'''18'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|323<br />
|-<br />
|95.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5526<br />
|-<br />
|29.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|93960<br />
|-<br />
|25.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1597338<br />
|-<br />
|10.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27154764<br />
|-<br />
|14.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|461631006<br />
|-<br />
|5.74%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7847727120<br />
|-<br />
|4.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|133411361058<br />
|-<br />
|4.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2267993138004<br />
|-<br />
|0.73%<br />
|}<br />
|-<br />
|'''19'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|360<br />
|-<br />
|93.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6517<br />
|-<br />
|25.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117325<br />
|-<br />
|20.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2111869<br />
|-<br />
|10.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|38013661<br />
|-<br />
|10.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|684245917<br />
|-<br />
|5.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12316426525<br />
|-<br />
|4.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|221695677469<br />
|-<br />
|3.99%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3990522194461<br />
|-<br />
|0.45%<br />
|}<br />
|-<br />
|'''20'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|399<br />
|-<br />
|95.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7620<br />
|-<br />
|25.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|144800<br />
|-<br />
|24.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2751220<br />
|-<br />
|10.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52273200<br />
|-<br />
|17.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|993190820<br />
|-<br />
|5.6%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|18870625600<br />
|-<br />
|4.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|358541886420<br />
|-<br />
|3.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6812295842000<br />
|-<br />
|1.14%<br />
|}<br />
|}<br />
</center><br />
<br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: red; text-align: center;" | * || The Moore bound.<br />
|-<br />
|style="background-color: #ABCDEF; text-align: center;" | * || Upper bound introduced by A. Hoffman, R. Singleton, Bannai, E. and Ito, T. <br />
|-<br />
|style="background-color: #999900; text-align: center;" | * || Upper bound introduced by Leif Jorgensen.<br />
|-<br />
|style="background-color: #336666; text-align: center;" | * || Optimal graphs found by Buset and by Molodtsov. <br />
|-<br />
|style="background-color: #ffff00; text-align: center;" | * || Graphs shown optimal.<br />
|-<br />
|}<br />
</center><br />
<br />
==References==<br />
* 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]<br />
* Bannai, E.; Ito, T. (1981), "Regular graphs with excess one", Discrete Mathematics 37:147-158, doi:10.1016/0012-365X(81)90215-6.<br />
* 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.<br />
* 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].<br />
* 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.<br />
* Erd&ouml;s P; Fajtlowicz, S.; Hoffman A. J. (1980), "Maximum degree in graphs of diameter 2", Networks 10: 87-90.<br />
* 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]. <br />
* L. K. Jorgensen (1992), "Diameters of cubic graphs", Discrete Applied Mathematics 37/38: 347-351, doi:10.1016/0166-218X(92)90144-Y.<br />
* 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.<br />
* Kurosawa, K.; Tsujii, S. (1981), "Considerations on diameter of communication networks", Electronics and Communications in Japan 64A (4): 37-45.<br />
* Loz, E.; Širáň, J. (2008), "New record graphs in the degree-diameter problem", Australasian Journal of Combinatorics 41: 63–80.<br />
* 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.<br />
* 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.<br />
* 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.<br />
* 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.<br />
* 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].<br />
* Molodtsov, S. G. (2006), "Largest Graphs of Diameter 2 and Maximum Degree 6", Lecture Notes in Computer Science 4123: 853-857.<br />
* 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.<br />
* Pineda-Villavicencio, G.; Miller, M., "Complete characterization of graphs of maximum degree 3 and defect at most 4", submitted.<br />
* 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. <br />
* 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.<br />
* Brown, W. G. (1966) On graphs that do not contain a Thomsen graph. Canadian Mathematical Bulletin, 9, 281 - 285.<br />
<br />
==External links==<br />
* [http://www-mat.upc.es/grup_de_grafs/ Degree Diameter] online table.<br />
* [http://www.eyal.com.au/wiki/The_Degree/Diameter_Problem Eyal Loz's] Degree-Diameter problem page.<br />
* [http://isu.indstate.edu/ge/DD/index.html Geoffrey Exoo's] Degree-Diameter record graphs page.<br />
<br />
<br />
[[Category:The Degree/Diameter Problem]]</div>
Guillermo
http://combinatoricswiki.org/index.php?title=The_Degree_Diameter_Problem_for_General_Graphs&diff=644
The Degree Diameter Problem for General Graphs
2022-02-18T05:39:05Z
<p>Guillermo: /* Table of the orders of the largest known graphs for the undirected degree diameter problem */</p>
<hr />
<div>==Introduction==<br />
The '''degree/diameter problem for general graphs''' can be stated as follows:<br />
<br />
''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''.''<br />
<br />
In attempting to settle the values of ''N(d,k)'', research activities in this problem have follow the following two directions:<br />
<br />
*Increasing the lower bounds for ''N(d,k)'' by constructing ever larger graphs.<br />
<br />
* Lowering and/or setting upper bounds for ''N(d,k)'' by proving the non-existence of graphs<br />
whose order is close to the Moore bounds ''M(d,k)=(d(d-1)<sup>k</sup>-2)(d-2)<sup>-1</sup>''.<br />
<br />
==Increasing the lower bounds for ''N(d,k)''==<br />
<br />
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.<br />
<br />
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&nbsp;≤&nbsp;''d''&nbsp;≤&nbsp;20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;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.<br />
<br />
<br />
===Table of the orders of the largest known graphs for the undirected degree diameter problem===<br />
<br />
<center> <br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
| '''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 <br />
|-<br />
<br />
| '''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 <br />
|-<br />
| '''5''' ||style="background-color: blue;" | '''24''' ||style="background-color: #ffff00;" | 72 ||style="background-color: #ffff00;" | 212 ||style="background-color: #FF9900;" | 624 ||style="background-color: #7BB661;" | 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 <br />
|-<br />
| '''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: #7BB661;" | 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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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: #7BB661;" | 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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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<br />
|-<br />
| '''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: #7BB661;" | 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<br />
|-<br />
| '''13''' ||style="background-color: #ff99ff;" | 162 ||style="background-color: #FFF8DC;" | 856 ||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<br />
|-<br />
| '''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: #7BB661;" | 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<br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|}<br />
</center><br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="3" cellpadding="3" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: blue; text-align: center;" | * || Other non Moore but optimal graphs. <br />
|-<br />
|style="background-color: #999900; text-align: center;" | * || Graph found by J. Allwright.<br />
|-<br />
|style="background-color: #336666; text-align: center;" | * || Graph found by G. Wegner.<br />
|-<br />
|style="background-color: #ffff00; text-align: center;" | * || Graphs found by G. Exoo.<br />
|-<br />
|style="background-color: #FFF8DC; text-align: center;" | * || Graphs found by V. Pelekhaty. The adjacency list can be found here [[File:Pelekhaty-856-13-3.pdf]]<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #666633; text-align: center;" | * || Graphs found by J. Gómez. <br />
|-<br />
|style="background-color: #993300; text-align: center;" | * || Graph found by M. Mitjana and F. Comellas. This graph was also found independently by M. Sampels.<br />
|-<br />
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by C. Delorme.<br />
|-<br />
|style="background-color: #6666ff; text-align: center;" | * || Graphs found by C. Delorme and G. Farhi.<br />
|-<br />
|style="background-color: #187eac; text-align: center;" | * || Graphs found by E. Canale. (2012)<br />
|-<br />
|style="background-color: #3399cc; text-align: center;" | * || Graph found by J. C. Bermond, C. Delorme, and G. Farhi<br />
|-<br />
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by J. Gómez and M. A. Fiol.<br />
|-<br />
|style="background-color: #999999; text-align: center;" | * || Graphs found by J. Gómez, M. A. Fiol, and O. Serra.<br />
|-<br />
|style="background-color: #66cc66; text-align: center;" | * || Graph found by M.A. Fiol and J.L.A. Yebra.<br />
|-<br />
|style="background-color: #666666; text-align: center;" | * || Graph found by F. Comellas and J. Gómez.<br />
|-<br />
|style="background-color: #ddff00; text-align: center;" | * || Graph found by Jianxiang Chen.<br />
|-<br />
|style="background-color: #7BB661; 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.<br />
|-<br />
|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. <br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #aa8268; text-align: center;" | * || Graphs found by A. Rodriguez. (2012)<br />
|-<br />
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by M. Sampels.<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #ffcc99; text-align: center;" | * || Graph found by M. Conder.<br />
|-<br />
|style="background-color: #cc0033; text-align: center;" | * || Graphs found by Brown, W. G. (1966).<br />
|-<br />
|style="background-color: #99FF00; text-align: center;" | * || Graph found by M. Abas. (2017). More details are available in a paper by the author.<br />
|}<br />
</center><br />
<br />
==Lowering and/or setting upper bounds for ''N(d,k)''== <br />
<br />
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)-&delta;'' for small ''&delta;''. The parameter ''&delta;'' is called the defect. Such graphs are called ''(d,k,-&delta;)''-graphs.<br />
<br />
For ''&delta;=1'' the only ''(d,k,-1)''-graphs are the cycles on ''2k'' vertices. Erd&ouml;s, Fajtlowitcz and Hoffman, who proved the non-existence of ''(d,2,-1)''-graphs for ''d&ne;3''. Then, Bannai and Ito, and also<br />
independently, Kurosawa and Tsujii, proved the non-existence of ''(d,k,-1)''-graphs for ''d&ge;3'' and ''k&ge;3''.<br />
<br />
For ''&delta;=2'', the ''(2,k,-2)''-graphs are the cycles on ''2k-1''. Considering ''d&ge;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. <br />
<br />
When ''&delta;=2'', ''d&ge;3'' and ''k&ge;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&ge;4'' by Leif Jorgensen, the non-existence of ''(4,k,-2)''-graphs with ''k&ge;3'' by Mirka Miller and Rino Simanjuntak, some structural properties of ''(5,k,-2)''-graphs with ''k&ge;3'' by Guillermo Pineda-Villavicencio and Mirka Miller, the obtaining of several necessary conditions for the existence of ''(d,2,-2)''-graphs with ''d&ge;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.<br />
<br />
For the case of ''&delta;&ge;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&ge;2'' by Guillermo Pineda-Villavicencio and Mirka Miller by proving the non-existence of ''(3,k,-4)''-graphs with ''k&ge;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. <br />
<br />
===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===<br />
<br />
<center><br />
{| border="1"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
|'''3'''<br />
| align="center" |<br />
{| border="2" style="background:red;" <br />
|10<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|20<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#336666;" <br />
|38<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|92<br />
|-<br />
|76.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|188<br />
|-<br />
|70.21%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|380<br />
|-<br />
|51.57%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|764<br />
|-<br />
|43.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|1532<br />
|-<br />
|39.16%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|3068<br />
|-<br />
|40.74%<br />
|}<br />
|-<br />
|'''4'''<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|15<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52<br />
|-<br />
|78.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160<br />
|-<br />
|61.25%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484<br />
|-<br />
|75.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1456<br />
|-<br />
|50.82%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4372<br />
|-<br />
|30.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13120<br />
|-<br />
|24.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|39364<br />
|-<br />
|19.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|118096<br />
|-<br />
|14.99%<br />
|}<br />
|-<br />
|'''5'''<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|24<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|104<br />
|-<br />
|69.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|424<br />
|-<br />
|50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1704<br />
|-<br />
|36.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6824<br />
|-<br />
|40.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27304<br />
|-<br />
|20.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|109224<br />
|-<br />
|15.59%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|436904<br />
|-<br />
|13.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1747624<br />
|-<br />
|10.70%<br />
|}<br />
|-<br />
|'''6'''<br />
| align="center" |<br />
{| border="2" style="background:#336666;" <br />
|32<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|186<br />
|-<br />
|59.67%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|936<br />
|-<br />
|41.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4686<br />
|-<br />
|29.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23436<br />
|-<br />
|33.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117186<br />
|-<br />
|16.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|585936<br />
|-<br />
|13.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929686<br />
|-<br />
|10.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14648436<br />
|-<br />
|8.55%<br />
|}<br />
|-<br />
|'''7'''<br />
| align="center" |<br />
{| border="2" style="background:red;" <br />
|50<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|301<br />
|-<br />
|55.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1813<br />
|-<br />
|37.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|10885<br />
|-<br />
|25.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|65317<br />
|-<br />
|18.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|391909<br />
|-<br />
|13.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2351461<br />
|-<br />
|10.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14108773<br />
|-<br />
|8.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|84652645<br />
|-<br />
|7.09%<br />
|}<br />
|-<br />
|'''8'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|63<br />
|-<br />
|90.47%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|456<br />
|-<br />
|55.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3200<br />
|-<br />
|34.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|22408<br />
|-<br />
|22.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|156864<br />
|-<br />
|25.29%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1098056<br />
|-<br />
|11.94%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7686400<br />
|-<br />
|9.56%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53804808<br />
|-<br />
|7.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|376633664<br />
|-<br />
|6.61%<br />
|}<br />
|-<br />
|'''9'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|80<br />
|-<br />
|92.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|657<br />
|-<br />
|89.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5265<br />
|-<br />
|29.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42129<br />
|-<br />
|19.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|337041<br />
|-<br />
|22.51%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2696337<br />
|-<br />
|10.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|21570705<br />
|-<br />
|7.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|172565649<br />
|-<br />
|7.02%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1380525201<br />
|-<br />
|4.77%<br />
|}<br />
|-<br />
|'''10'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|99<br />
|-<br />
|91.91%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|910<br />
|-<br />
|71.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8200<br />
|-<br />
|27.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73810<br />
|-<br />
|17.80%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|664300<br />
|-<br />
|20.27%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5978710<br />
|-<br />
|9.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53808400<br />
|-<br />
|7.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484275610<br />
|-<br />
|5.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4358480500<br />
|-<br />
|4.61%<br />
|}<br />
|-<br />
|'''11'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|120<br />
|-<br />
|86.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1221<br />
|-<br />
|58.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12221<br />
|-<br />
|26.18%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122221<br />
|-<br />
|15.95%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222221<br />
|-<br />
|12.83%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222221<br />
|-<br />
|8.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122222221<br />
|-<br />
|6.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222222221<br />
|-<br />
|5.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222222221<br />
|-<br />
|4.91%<br />
|}<br />
|-<br />
|'''12'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|143<br />
|-<br />
|93%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1596<br />
|-<br />
|49.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|17568<br />
|-<br />
|26.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|193260<br />
|-<br />
|15.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2125872<br />
|-<br />
|16.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23384604<br />
|-<br />
|8.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|257230656<br />
|-<br />
|6.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2829537228<br />
|-<br />
|5.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|31124909520<br />
|-<br />
|4.83%<br />
|}<br />
|-<br />
|'''13'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|168<br />
|-<br />
|96.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2041<br />
|-<br />
|41.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|24505<br />
|-<br />
|26.77%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|294073<br />
|-<br />
|13.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3528889<br />
|-<br />
|15.05%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42346681<br />
|-<br />
|7.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|508160185<br />
|-<br />
|5.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6097922233<br />
|-<br />
|4.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73175066809<br />
|-<br />
|4.20%<br />
|}<br />
|-<br />
|'''14'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195<br />
|-<br />
|93.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2562<br />
|-<br />
|35.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|33320<br />
|-<br />
|24.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|433174<br />
|-<br />
|13.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5631276<br />
|-<br />
|14.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73206602<br />
|-<br />
|8.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|951685840<br />
|-<br />
|5.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12371915934<br />
|-<br />
|4.85%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160834907156<br />
|-<br />
|4.37%<br />
|}<br />
|-<br />
|'''15'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|224<br />
|-<br />
|83.03%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3165<br />
|-<br />
|38.38%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|44325<br />
|-<br />
|26.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|620565<br />
|-<br />
|12.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8687925<br />
|-<br />
|16.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|121630965<br />
|-<br />
|7.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1702833525<br />
|-<br />
|5.28%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23839669365<br />
|-<br />
|4.91%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|333755371125<br />
|-<br />
|2.99%<br />
|}<br />
|-<br />
|'''16'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|255<br />
|-<br />
|77.64%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3856<br />
|-<br />
|41.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|57856<br />
|-<br />
|25.30%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|867856<br />
|-<br />
|15.26%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13017856<br />
|-<br />
|13.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195267856<br />
|-<br />
|7.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929017856<br />
|-<br />
|4.79%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|43935267856<br />
|-<br />
|4.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|659029017856<br />
|-<br />
|1.96%<br />
|}<br />
|-<br />
|'''17'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|288<br />
|-<br />
|95.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4641<br />
|-<br />
|34.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|74273<br />
|-<br />
|25.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1188385<br />
|-<br />
|11.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|19014177<br />
|-<br />
|16.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|304226849<br />
|-<br />
|6.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4867629601<br />
|-<br />
|4.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|77882073633<br />
|-<br />
|4.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1246113178145<br />
|-<br />
|1.22%<br />
|}<br />
|-<br />
|'''18'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|323<br />
|-<br />
|95.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5526<br />
|-<br />
|29.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|93960<br />
|-<br />
|25.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1597338<br />
|-<br />
|10.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27154764<br />
|-<br />
|14.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|461631006<br />
|-<br />
|5.74%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7847727120<br />
|-<br />
|4.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|133411361058<br />
|-<br />
|4.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2267993138004<br />
|-<br />
|0.73%<br />
|}<br />
|-<br />
|'''19'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|360<br />
|-<br />
|93.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6517<br />
|-<br />
|25.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117325<br />
|-<br />
|20.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2111869<br />
|-<br />
|10.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|38013661<br />
|-<br />
|10.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|684245917<br />
|-<br />
|5.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12316426525<br />
|-<br />
|4.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|221695677469<br />
|-<br />
|3.99%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3990522194461<br />
|-<br />
|0.45%<br />
|}<br />
|-<br />
|'''20'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|399<br />
|-<br />
|95.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7620<br />
|-<br />
|25.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|144800<br />
|-<br />
|24.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2751220<br />
|-<br />
|10.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52273200<br />
|-<br />
|17.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|993190820<br />
|-<br />
|5.6%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|18870625600<br />
|-<br />
|4.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|358541886420<br />
|-<br />
|3.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6812295842000<br />
|-<br />
|1.14%<br />
|}<br />
|}<br />
</center><br />
<br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: red; text-align: center;" | * || The Moore bound.<br />
|-<br />
|style="background-color: #ABCDEF; text-align: center;" | * || Upper bound introduced by A. Hoffman, R. Singleton, Bannai, E. and Ito, T. <br />
|-<br />
|style="background-color: #999900; text-align: center;" | * || Upper bound introduced by Leif Jorgensen.<br />
|-<br />
|style="background-color: #336666; text-align: center;" | * || Optimal graphs found by Buset and by Molodtsov. <br />
|-<br />
|style="background-color: #ffff00; text-align: center;" | * || Graphs shown optimal.<br />
|-<br />
|}<br />
</center><br />
<br />
==References==<br />
* 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]<br />
* Bannai, E.; Ito, T. (1981), "Regular graphs with excess one", Discrete Mathematics 37:147-158, doi:10.1016/0012-365X(81)90215-6.<br />
* 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.<br />
* 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].<br />
* 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.<br />
* Erd&ouml;s P; Fajtlowicz, S.; Hoffman A. J. (1980), "Maximum degree in graphs of diameter 2", Networks 10: 87-90.<br />
* 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]. <br />
* L. K. Jorgensen (1992), "Diameters of cubic graphs", Discrete Applied Mathematics 37/38: 347-351, doi:10.1016/0166-218X(92)90144-Y.<br />
* 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.<br />
* Kurosawa, K.; Tsujii, S. (1981), "Considerations on diameter of communication networks", Electronics and Communications in Japan 64A (4): 37-45.<br />
* Loz, E.; Širáň, J. (2008), "New record graphs in the degree-diameter problem", Australasian Journal of Combinatorics 41: 63–80.<br />
* 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.<br />
* 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.<br />
* 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.<br />
* 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.<br />
* 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].<br />
* Molodtsov, S. G. (2006), "Largest Graphs of Diameter 2 and Maximum Degree 6", Lecture Notes in Computer Science 4123: 853-857.<br />
* 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.<br />
* Pineda-Villavicencio, G.; Miller, M., "Complete characterization of graphs of maximum degree 3 and defect at most 4", submitted.<br />
* 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. <br />
* 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.<br />
* Brown, W. G. (1966) On graphs that do not contain a Thomsen graph. Canadian Mathematical Bulletin, 9, 281 - 285.<br />
<br />
==External links==<br />
* [http://www-mat.upc.es/grup_de_grafs/ Degree Diameter] online table.<br />
* [http://www.eyal.com.au/wiki/The_Degree/Diameter_Problem Eyal Loz's] Degree-Diameter problem page.<br />
* [http://isu.indstate.edu/ge/DD/index.html Geoffrey Exoo's] Degree-Diameter record graphs page.<br />
<br />
<br />
[[Category:The Degree/Diameter Problem]]</div>
Guillermo
http://combinatoricswiki.org/index.php?title=File:Pelekhaty-856-13-3.pdf&diff=643
File:Pelekhaty-856-13-3.pdf
2022-02-18T05:35:14Z
<p>Guillermo: </p>
<hr />
<div></div>
Guillermo
http://combinatoricswiki.org/index.php?title=The_Degree_Diameter_Problem_for_General_Graphs&diff=642
The Degree Diameter Problem for General Graphs
2022-02-18T05:14:06Z
<p>Guillermo: /* Table of the orders of the largest known graphs for the undirected degree diameter problem */</p>
<hr />
<div>==Introduction==<br />
The '''degree/diameter problem for general graphs''' can be stated as follows:<br />
<br />
''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''.''<br />
<br />
In attempting to settle the values of ''N(d,k)'', research activities in this problem have follow the following two directions:<br />
<br />
*Increasing the lower bounds for ''N(d,k)'' by constructing ever larger graphs.<br />
<br />
* Lowering and/or setting upper bounds for ''N(d,k)'' by proving the non-existence of graphs<br />
whose order is close to the Moore bounds ''M(d,k)=(d(d-1)<sup>k</sup>-2)(d-2)<sup>-1</sup>''.<br />
<br />
==Increasing the lower bounds for ''N(d,k)''==<br />
<br />
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.<br />
<br />
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&nbsp;≤&nbsp;''d''&nbsp;≤&nbsp;20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;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.<br />
<br />
<br />
===Table of the orders of the largest known graphs for the undirected degree diameter problem===<br />
<br />
<center> <br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
| '''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 <br />
|-<br />
<br />
| '''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 <br />
|-<br />
| '''5''' ||style="background-color: blue;" | '''24''' ||style="background-color: #ffff00;" | 72 ||style="background-color: #ffff00;" | 212 ||style="background-color: #FF9900;" | 624 ||style="background-color: #7BB661;" | 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 <br />
|-<br />
| '''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: #7BB661;" | 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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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: #7BB661;" | 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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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<br />
|-<br />
| '''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: #7BB661;" | 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<br />
|-<br />
| '''13''' ||style="background-color: #ff99ff;" | 162 ||style="background-color: #FFF8DC;" | 856 ||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<br />
|-<br />
| '''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: #7BB661;" | 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<br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|}<br />
</center><br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="3" cellpadding="3" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: blue; text-align: center;" | * || Other non Moore but optimal graphs. <br />
|-<br />
|style="background-color: #999900; text-align: center;" | * || Graph found by J. Allwright.<br />
|-<br />
|style="background-color: #336666; text-align: center;" | * || Graph found by G. Wegner.<br />
|-<br />
|style="background-color: #ffff00; text-align: center;" | * || Graphs found by G. Exoo.<br />
|-<br />
|style="background-color: #FFF8DC; text-align: center;" | * || Graphs found by V. Pelekhaty.<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #666633; text-align: center;" | * || Graphs found by J. Gómez. <br />
|-<br />
|style="background-color: #993300; text-align: center;" | * || Graph found by M. Mitjana and F. Comellas. This graph was also found independently by M. Sampels.<br />
|-<br />
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by C. Delorme.<br />
|-<br />
|style="background-color: #6666ff; text-align: center;" | * || Graphs found by C. Delorme and G. Farhi.<br />
|-<br />
|style="background-color: #187eac; text-align: center;" | * || Graphs found by E. Canale. (2012)<br />
|-<br />
|style="background-color: #3399cc; text-align: center;" | * || Graph found by J. C. Bermond, C. Delorme, and G. Farhi<br />
|-<br />
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by J. Gómez and M. A. Fiol.<br />
|-<br />
|style="background-color: #999999; text-align: center;" | * || Graphs found by J. Gómez, M. A. Fiol, and O. Serra.<br />
|-<br />
|style="background-color: #66cc66; text-align: center;" | * || Graph found by M.A. Fiol and J.L.A. Yebra.<br />
|-<br />
|style="background-color: #666666; text-align: center;" | * || Graph found by F. Comellas and J. Gómez.<br />
|-<br />
|style="background-color: #ddff00; text-align: center;" | * || Graph found by Jianxiang Chen.<br />
|-<br />
|style="background-color: #7BB661; 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.<br />
|-<br />
|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. <br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #aa8268; text-align: center;" | * || Graphs found by A. Rodriguez. (2012)<br />
|-<br />
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by M. Sampels.<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #ffcc99; text-align: center;" | * || Graph found by M. Conder.<br />
|-<br />
|style="background-color: #cc0033; text-align: center;" | * || Graphs found by Brown, W. G. (1966).<br />
|-<br />
|style="background-color: #99FF00; text-align: center;" | * || Graph found by M. Abas. (2017). More details are available in a paper by the author.<br />
|}<br />
</center><br />
<br />
==Lowering and/or setting upper bounds for ''N(d,k)''== <br />
<br />
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)-&delta;'' for small ''&delta;''. The parameter ''&delta;'' is called the defect. Such graphs are called ''(d,k,-&delta;)''-graphs.<br />
<br />
For ''&delta;=1'' the only ''(d,k,-1)''-graphs are the cycles on ''2k'' vertices. Erd&ouml;s, Fajtlowitcz and Hoffman, who proved the non-existence of ''(d,2,-1)''-graphs for ''d&ne;3''. Then, Bannai and Ito, and also<br />
independently, Kurosawa and Tsujii, proved the non-existence of ''(d,k,-1)''-graphs for ''d&ge;3'' and ''k&ge;3''.<br />
<br />
For ''&delta;=2'', the ''(2,k,-2)''-graphs are the cycles on ''2k-1''. Considering ''d&ge;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. <br />
<br />
When ''&delta;=2'', ''d&ge;3'' and ''k&ge;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&ge;4'' by Leif Jorgensen, the non-existence of ''(4,k,-2)''-graphs with ''k&ge;3'' by Mirka Miller and Rino Simanjuntak, some structural properties of ''(5,k,-2)''-graphs with ''k&ge;3'' by Guillermo Pineda-Villavicencio and Mirka Miller, the obtaining of several necessary conditions for the existence of ''(d,2,-2)''-graphs with ''d&ge;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.<br />
<br />
For the case of ''&delta;&ge;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&ge;2'' by Guillermo Pineda-Villavicencio and Mirka Miller by proving the non-existence of ''(3,k,-4)''-graphs with ''k&ge;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. <br />
<br />
===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===<br />
<br />
<center><br />
{| border="1"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
|'''3'''<br />
| align="center" |<br />
{| border="2" style="background:red;" <br />
|10<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|20<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#336666;" <br />
|38<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|92<br />
|-<br />
|76.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|188<br />
|-<br />
|70.21%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|380<br />
|-<br />
|51.57%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|764<br />
|-<br />
|43.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|1532<br />
|-<br />
|39.16%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|3068<br />
|-<br />
|40.74%<br />
|}<br />
|-<br />
|'''4'''<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|15<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52<br />
|-<br />
|78.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160<br />
|-<br />
|61.25%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484<br />
|-<br />
|75.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1456<br />
|-<br />
|50.82%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4372<br />
|-<br />
|30.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13120<br />
|-<br />
|24.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|39364<br />
|-<br />
|19.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|118096<br />
|-<br />
|14.99%<br />
|}<br />
|-<br />
|'''5'''<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|24<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|104<br />
|-<br />
|69.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|424<br />
|-<br />
|50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1704<br />
|-<br />
|36.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6824<br />
|-<br />
|40.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27304<br />
|-<br />
|20.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|109224<br />
|-<br />
|15.59%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|436904<br />
|-<br />
|13.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1747624<br />
|-<br />
|10.70%<br />
|}<br />
|-<br />
|'''6'''<br />
| align="center" |<br />
{| border="2" style="background:#336666;" <br />
|32<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|186<br />
|-<br />
|59.67%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|936<br />
|-<br />
|41.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4686<br />
|-<br />
|29.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23436<br />
|-<br />
|33.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117186<br />
|-<br />
|16.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|585936<br />
|-<br />
|13.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929686<br />
|-<br />
|10.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14648436<br />
|-<br />
|8.55%<br />
|}<br />
|-<br />
|'''7'''<br />
| align="center" |<br />
{| border="2" style="background:red;" <br />
|50<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|301<br />
|-<br />
|55.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1813<br />
|-<br />
|37.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|10885<br />
|-<br />
|25.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|65317<br />
|-<br />
|18.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|391909<br />
|-<br />
|13.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2351461<br />
|-<br />
|10.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14108773<br />
|-<br />
|8.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|84652645<br />
|-<br />
|7.09%<br />
|}<br />
|-<br />
|'''8'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|63<br />
|-<br />
|90.47%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|456<br />
|-<br />
|55.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3200<br />
|-<br />
|34.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|22408<br />
|-<br />
|22.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|156864<br />
|-<br />
|25.29%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1098056<br />
|-<br />
|11.94%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7686400<br />
|-<br />
|9.56%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53804808<br />
|-<br />
|7.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|376633664<br />
|-<br />
|6.61%<br />
|}<br />
|-<br />
|'''9'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|80<br />
|-<br />
|92.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|657<br />
|-<br />
|89.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5265<br />
|-<br />
|29.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42129<br />
|-<br />
|19.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|337041<br />
|-<br />
|22.51%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2696337<br />
|-<br />
|10.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|21570705<br />
|-<br />
|7.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|172565649<br />
|-<br />
|7.02%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1380525201<br />
|-<br />
|4.77%<br />
|}<br />
|-<br />
|'''10'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|99<br />
|-<br />
|91.91%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|910<br />
|-<br />
|71.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8200<br />
|-<br />
|27.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73810<br />
|-<br />
|17.80%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|664300<br />
|-<br />
|20.27%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5978710<br />
|-<br />
|9.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53808400<br />
|-<br />
|7.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484275610<br />
|-<br />
|5.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4358480500<br />
|-<br />
|4.61%<br />
|}<br />
|-<br />
|'''11'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|120<br />
|-<br />
|86.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1221<br />
|-<br />
|58.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12221<br />
|-<br />
|26.18%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122221<br />
|-<br />
|15.95%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222221<br />
|-<br />
|12.83%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222221<br />
|-<br />
|8.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122222221<br />
|-<br />
|6.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222222221<br />
|-<br />
|5.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222222221<br />
|-<br />
|4.91%<br />
|}<br />
|-<br />
|'''12'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|143<br />
|-<br />
|93%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1596<br />
|-<br />
|49.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|17568<br />
|-<br />
|26.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|193260<br />
|-<br />
|15.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2125872<br />
|-<br />
|16.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23384604<br />
|-<br />
|8.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|257230656<br />
|-<br />
|6.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2829537228<br />
|-<br />
|5.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|31124909520<br />
|-<br />
|4.83%<br />
|}<br />
|-<br />
|'''13'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|168<br />
|-<br />
|96.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2041<br />
|-<br />
|41.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|24505<br />
|-<br />
|26.77%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|294073<br />
|-<br />
|13.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3528889<br />
|-<br />
|15.05%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42346681<br />
|-<br />
|7.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|508160185<br />
|-<br />
|5.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6097922233<br />
|-<br />
|4.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73175066809<br />
|-<br />
|4.20%<br />
|}<br />
|-<br />
|'''14'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195<br />
|-<br />
|93.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2562<br />
|-<br />
|35.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|33320<br />
|-<br />
|24.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|433174<br />
|-<br />
|13.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5631276<br />
|-<br />
|14.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73206602<br />
|-<br />
|8.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|951685840<br />
|-<br />
|5.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12371915934<br />
|-<br />
|4.85%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160834907156<br />
|-<br />
|4.37%<br />
|}<br />
|-<br />
|'''15'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|224<br />
|-<br />
|83.03%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3165<br />
|-<br />
|38.38%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|44325<br />
|-<br />
|26.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|620565<br />
|-<br />
|12.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8687925<br />
|-<br />
|16.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|121630965<br />
|-<br />
|7.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1702833525<br />
|-<br />
|5.28%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23839669365<br />
|-<br />
|4.91%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|333755371125<br />
|-<br />
|2.99%<br />
|}<br />
|-<br />
|'''16'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|255<br />
|-<br />
|77.64%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3856<br />
|-<br />
|41.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|57856<br />
|-<br />
|25.30%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|867856<br />
|-<br />
|15.26%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13017856<br />
|-<br />
|13.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195267856<br />
|-<br />
|7.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929017856<br />
|-<br />
|4.79%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|43935267856<br />
|-<br />
|4.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|659029017856<br />
|-<br />
|1.96%<br />
|}<br />
|-<br />
|'''17'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|288<br />
|-<br />
|95.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4641<br />
|-<br />
|34.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|74273<br />
|-<br />
|25.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1188385<br />
|-<br />
|11.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|19014177<br />
|-<br />
|16.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|304226849<br />
|-<br />
|6.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4867629601<br />
|-<br />
|4.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|77882073633<br />
|-<br />
|4.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1246113178145<br />
|-<br />
|1.22%<br />
|}<br />
|-<br />
|'''18'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|323<br />
|-<br />
|95.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5526<br />
|-<br />
|29.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|93960<br />
|-<br />
|25.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1597338<br />
|-<br />
|10.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27154764<br />
|-<br />
|14.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|461631006<br />
|-<br />
|5.74%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7847727120<br />
|-<br />
|4.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|133411361058<br />
|-<br />
|4.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2267993138004<br />
|-<br />
|0.73%<br />
|}<br />
|-<br />
|'''19'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|360<br />
|-<br />
|93.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6517<br />
|-<br />
|25.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117325<br />
|-<br />
|20.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2111869<br />
|-<br />
|10.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|38013661<br />
|-<br />
|10.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|684245917<br />
|-<br />
|5.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12316426525<br />
|-<br />
|4.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|221695677469<br />
|-<br />
|3.99%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3990522194461<br />
|-<br />
|0.45%<br />
|}<br />
|-<br />
|'''20'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|399<br />
|-<br />
|95.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7620<br />
|-<br />
|25.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|144800<br />
|-<br />
|24.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2751220<br />
|-<br />
|10.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52273200<br />
|-<br />
|17.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|993190820<br />
|-<br />
|5.6%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|18870625600<br />
|-<br />
|4.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|358541886420<br />
|-<br />
|3.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6812295842000<br />
|-<br />
|1.14%<br />
|}<br />
|}<br />
</center><br />
<br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: red; text-align: center;" | * || The Moore bound.<br />
|-<br />
|style="background-color: #ABCDEF; text-align: center;" | * || Upper bound introduced by A. Hoffman, R. Singleton, Bannai, E. and Ito, T. <br />
|-<br />
|style="background-color: #999900; text-align: center;" | * || Upper bound introduced by Leif Jorgensen.<br />
|-<br />
|style="background-color: #336666; text-align: center;" | * || Optimal graphs found by Buset and by Molodtsov. <br />
|-<br />
|style="background-color: #ffff00; text-align: center;" | * || Graphs shown optimal.<br />
|-<br />
|}<br />
</center><br />
<br />
==References==<br />
* 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]<br />
* Bannai, E.; Ito, T. (1981), "Regular graphs with excess one", Discrete Mathematics 37:147-158, doi:10.1016/0012-365X(81)90215-6.<br />
* 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.<br />
* 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].<br />
* 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.<br />
* Erd&ouml;s P; Fajtlowicz, S.; Hoffman A. J. (1980), "Maximum degree in graphs of diameter 2", Networks 10: 87-90.<br />
* 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]. <br />
* L. K. Jorgensen (1992), "Diameters of cubic graphs", Discrete Applied Mathematics 37/38: 347-351, doi:10.1016/0166-218X(92)90144-Y.<br />
* 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.<br />
* Kurosawa, K.; Tsujii, S. (1981), "Considerations on diameter of communication networks", Electronics and Communications in Japan 64A (4): 37-45.<br />
* Loz, E.; Širáň, J. (2008), "New record graphs in the degree-diameter problem", Australasian Journal of Combinatorics 41: 63–80.<br />
* 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.<br />
* 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.<br />
* 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.<br />
* 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.<br />
* 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].<br />
* Molodtsov, S. G. (2006), "Largest Graphs of Diameter 2 and Maximum Degree 6", Lecture Notes in Computer Science 4123: 853-857.<br />
* 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.<br />
* Pineda-Villavicencio, G.; Miller, M., "Complete characterization of graphs of maximum degree 3 and defect at most 4", submitted.<br />
* 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. <br />
* 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.<br />
* Brown, W. G. (1966) On graphs that do not contain a Thomsen graph. Canadian Mathematical Bulletin, 9, 281 - 285.<br />
<br />
==External links==<br />
* [http://www-mat.upc.es/grup_de_grafs/ Degree Diameter] online table.<br />
* [http://www.eyal.com.au/wiki/The_Degree/Diameter_Problem Eyal Loz's] Degree-Diameter problem page.<br />
* [http://isu.indstate.edu/ge/DD/index.html Geoffrey Exoo's] Degree-Diameter record graphs page.<br />
<br />
<br />
[[Category:The Degree/Diameter Problem]]</div>
Guillermo
http://combinatoricswiki.org/index.php?title=The_Degree_Diameter_Problem_for_General_Graphs&diff=641
The Degree Diameter Problem for General Graphs
2022-02-18T04:06:12Z
<p>Guillermo: /* Table of the orders of the largest known graphs for the undirected degree diameter problem */</p>
<hr />
<div>==Introduction==<br />
The '''degree/diameter problem for general graphs''' can be stated as follows:<br />
<br />
''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''.''<br />
<br />
In attempting to settle the values of ''N(d,k)'', research activities in this problem have follow the following two directions:<br />
<br />
*Increasing the lower bounds for ''N(d,k)'' by constructing ever larger graphs.<br />
<br />
* Lowering and/or setting upper bounds for ''N(d,k)'' by proving the non-existence of graphs<br />
whose order is close to the Moore bounds ''M(d,k)=(d(d-1)<sup>k</sup>-2)(d-2)<sup>-1</sup>''.<br />
<br />
==Increasing the lower bounds for ''N(d,k)''==<br />
<br />
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.<br />
<br />
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&nbsp;≤&nbsp;''d''&nbsp;≤&nbsp;20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;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.<br />
<br />
<br />
===Table of the orders of the largest known graphs for the undirected degree diameter problem===<br />
<br />
<center> <br />
{| border="1" cellspacing="2" cellpadding="2" style="text-align: center;"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
| '''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 <br />
|-<br />
<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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 <br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
|-<br />
| '''13''' ||style="background-color: #ff99ff;" | 162 ||style="background-color: #666633;" | 856 ||style="background-color: #CAE00D; 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<br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|-<br />
| '''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<br />
<br />
|}<br />
</center><br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: blue; text-align: center;" | * || Other non Moore but optimal graphs. <br />
|-<br />
|style="background-color: #999900; text-align: center;" | * || Graph found by J. Allwright.<br />
|-<br />
|style="background-color: #336666; text-align: center;" | * || Graph found by G. Wegner.<br />
|-<br />
|style="background-color: #ffff00; text-align: center;" | * || Graphs found by G. Exoo.<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #666633; text-align: center;" | * || Graphs found by J. Gómez. <br />
|-<br />
|style="background-color: #993300; text-align: center;" | * || Graph found by M. Mitjana and F. Comellas. This graph was also found independently by M. Sampels.<br />
|-<br />
|style="background-color: #81BEF7; text-align: center;" | * || Graphs found by C. Delorme.<br />
|-<br />
|style="background-color: #6666ff; text-align: center;" | * || Graphs found by C. Delorme and G. Farhi.<br />
|-<br />
|style="background-color: #187eac; text-align: center;" | * || Graphs found by E. Canale. (2012)<br />
|-<br />
|style="background-color: #3399cc; text-align: center;" | * || Graph found by J. C. Bermond, C. Delorme, and G. Farhi<br />
|-<br />
|style="background-color: #cccccc; text-align: center;" | * || Graphs found by J. Gómez and M. A. Fiol.<br />
|-<br />
|style="background-color: #999999; text-align: center;" | * || Graphs found by J. Gómez, M. A. Fiol, and O. Serra.<br />
|-<br />
|style="background-color: #66cc66; text-align: center;" | * || Graph found by M.A. Fiol and J.L.A. Yebra.<br />
|-<br />
|style="background-color: #666666; text-align: center;" | * || Graph found by F. Comellas and J. Gómez.<br />
|-<br />
|style="background-color: #ddff00; text-align: center;" | * || Graph found by Jianxiang Chen.<br />
|-<br />
|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.<br />
|-<br />
|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. <br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #aa8268; text-align: center;" | * || Graphs found by A. Rodriguez. (2012)<br />
|-<br />
|style="background-color: #bbffff; text-align: center;" | * || Graphs found by M. Sampels.<br />
|-<br />
|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.<br />
|-<br />
|style="background-color: #ffcc99; text-align: center;" | * || Graph found by M. Conder.<br />
|-<br />
|style="background-color: #cc0033; text-align: center;" | * || Graphs found by Brown, W. G. (1966).<br />
|-<br />
|style="background-color: #99FF00; text-align: center;" | * || Graph found by M. Abas. (2017). More details are available in a paper by the author.<br />
|}<br />
</center><br />
<br />
==Lowering and/or setting upper bounds for ''N(d,k)''== <br />
<br />
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)-&delta;'' for small ''&delta;''. The parameter ''&delta;'' is called the defect. Such graphs are called ''(d,k,-&delta;)''-graphs.<br />
<br />
For ''&delta;=1'' the only ''(d,k,-1)''-graphs are the cycles on ''2k'' vertices. Erd&ouml;s, Fajtlowitcz and Hoffman, who proved the non-existence of ''(d,2,-1)''-graphs for ''d&ne;3''. Then, Bannai and Ito, and also<br />
independently, Kurosawa and Tsujii, proved the non-existence of ''(d,k,-1)''-graphs for ''d&ge;3'' and ''k&ge;3''.<br />
<br />
For ''&delta;=2'', the ''(2,k,-2)''-graphs are the cycles on ''2k-1''. Considering ''d&ge;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. <br />
<br />
When ''&delta;=2'', ''d&ge;3'' and ''k&ge;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&ge;4'' by Leif Jorgensen, the non-existence of ''(4,k,-2)''-graphs with ''k&ge;3'' by Mirka Miller and Rino Simanjuntak, some structural properties of ''(5,k,-2)''-graphs with ''k&ge;3'' by Guillermo Pineda-Villavicencio and Mirka Miller, the obtaining of several necessary conditions for the existence of ''(d,2,-2)''-graphs with ''d&ge;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.<br />
<br />
For the case of ''&delta;&ge;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&ge;2'' by Guillermo Pineda-Villavicencio and Mirka Miller by proving the non-existence of ''(3,k,-4)''-graphs with ''k&ge;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. <br />
<br />
===Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs===<br />
<br />
<center><br />
{| border="1"<br />
| '''<math>d</math>\<math>k</math>'''|| '''2''' || '''3''' || '''4'''|| '''5''' || '''6''' || '''7''' || '''8''' || '''9''' || '''10''' <br />
|-<br />
|'''3'''<br />
| align="center" |<br />
{| border="2" style="background:red;" <br />
|10<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|20<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#336666;" <br />
|38<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|92<br />
|-<br />
|76.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|188<br />
|-<br />
|70.21%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|380<br />
|-<br />
|51.57%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|764<br />
|-<br />
|43.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|1532<br />
|-<br />
|39.16%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#999900;" <br />
|3068<br />
|-<br />
|40.74%<br />
|}<br />
|-<br />
|'''4'''<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|15<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52<br />
|-<br />
|78.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160<br />
|-<br />
|61.25%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484<br />
|-<br />
|75.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1456<br />
|-<br />
|50.82%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4372<br />
|-<br />
|30.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13120<br />
|-<br />
|24.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|39364<br />
|-<br />
|19.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|118096<br />
|-<br />
|14.99%<br />
|}<br />
|-<br />
|'''5'''<br />
| align="center" |<br />
{| border="2" style="background:#ffff00;" <br />
|24<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|104<br />
|-<br />
|69.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|424<br />
|-<br />
|50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1704<br />
|-<br />
|36.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6824<br />
|-<br />
|40.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27304<br />
|-<br />
|20.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|109224<br />
|-<br />
|15.59%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|436904<br />
|-<br />
|13.23%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1747624<br />
|-<br />
|10.70%<br />
|}<br />
|-<br />
|'''6'''<br />
| align="center" |<br />
{| border="2" style="background:#336666;" <br />
|32<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|186<br />
|-<br />
|59.67%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|936<br />
|-<br />
|41.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4686<br />
|-<br />
|29.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23436<br />
|-<br />
|33.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117186<br />
|-<br />
|16.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|585936<br />
|-<br />
|13.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929686<br />
|-<br />
|10.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14648436<br />
|-<br />
|8.55%<br />
|}<br />
|-<br />
|'''7'''<br />
| align="center" |<br />
{| border="2" style="background:red;" <br />
|50<br />
|-<br />
|100%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|301<br />
|-<br />
|55.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1813<br />
|-<br />
|37.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|10885<br />
|-<br />
|25.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|65317<br />
|-<br />
|18.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|391909<br />
|-<br />
|13.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2351461<br />
|-<br />
|10.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|14108773<br />
|-<br />
|8.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|84652645<br />
|-<br />
|7.09%<br />
|}<br />
|-<br />
|'''8'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|63<br />
|-<br />
|90.47%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|456<br />
|-<br />
|55.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3200<br />
|-<br />
|34.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|22408<br />
|-<br />
|22.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|156864<br />
|-<br />
|25.29%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1098056<br />
|-<br />
|11.94%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7686400<br />
|-<br />
|9.56%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53804808<br />
|-<br />
|7.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|376633664<br />
|-<br />
|6.61%<br />
|}<br />
|-<br />
|'''9'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|80<br />
|-<br />
|92.50%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|657<br />
|-<br />
|89.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5265<br />
|-<br />
|29.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42129<br />
|-<br />
|19.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|337041<br />
|-<br />
|22.51%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2696337<br />
|-<br />
|10.37%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|21570705<br />
|-<br />
|7.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|172565649<br />
|-<br />
|7.02%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1380525201<br />
|-<br />
|4.77%<br />
|}<br />
|-<br />
|'''10'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|99<br />
|-<br />
|91.91%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|910<br />
|-<br />
|71.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8200<br />
|-<br />
|27.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73810<br />
|-<br />
|17.80%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|664300<br />
|-<br />
|20.27%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5978710<br />
|-<br />
|9.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|53808400<br />
|-<br />
|7.97%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|484275610<br />
|-<br />
|5.78%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4358480500<br />
|-<br />
|4.61%<br />
|}<br />
|-<br />
|'''11'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|120<br />
|-<br />
|86.66%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1221<br />
|-<br />
|58.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12221<br />
|-<br />
|26.18%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122221<br />
|-<br />
|15.95%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222221<br />
|-<br />
|12.83%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222221<br />
|-<br />
|8.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|122222221<br />
|-<br />
|6.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1222222221<br />
|-<br />
|5.96%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12222222221<br />
|-<br />
|4.91%<br />
|}<br />
|-<br />
|'''12'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|143<br />
|-<br />
|93%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1596<br />
|-<br />
|49.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|17568<br />
|-<br />
|26.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|193260<br />
|-<br />
|15.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2125872<br />
|-<br />
|16.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23384604<br />
|-<br />
|8.55%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|257230656<br />
|-<br />
|6.19%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2829537228<br />
|-<br />
|5.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|31124909520<br />
|-<br />
|4.83%<br />
|}<br />
|-<br />
|'''13'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|168<br />
|-<br />
|96.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2041<br />
|-<br />
|41.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|24505<br />
|-<br />
|26.77%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|294073<br />
|-<br />
|13.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3528889<br />
|-<br />
|15.05%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|42346681<br />
|-<br />
|7.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|508160185<br />
|-<br />
|5.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6097922233<br />
|-<br />
|4.08%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73175066809<br />
|-<br />
|4.20%<br />
|}<br />
|-<br />
|'''14'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195<br />
|-<br />
|93.84%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2562<br />
|-<br />
|35.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|33320<br />
|-<br />
|24.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|433174<br />
|-<br />
|13.35%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5631276<br />
|-<br />
|14.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|73206602<br />
|-<br />
|8.46%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|951685840<br />
|-<br />
|5.87%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12371915934<br />
|-<br />
|4.85%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|160834907156<br />
|-<br />
|4.37%<br />
|}<br />
|-<br />
|'''15'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|224<br />
|-<br />
|83.03%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3165<br />
|-<br />
|38.38%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|44325<br />
|-<br />
|26.42%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|620565<br />
|-<br />
|12.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|8687925<br />
|-<br />
|16.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|121630965<br />
|-<br />
|7.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1702833525<br />
|-<br />
|5.28%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|23839669365<br />
|-<br />
|4.91%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|333755371125<br />
|-<br />
|2.99%<br />
|}<br />
|-<br />
|'''16'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|255<br />
|-<br />
|77.64%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3856<br />
|-<br />
|41.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|57856<br />
|-<br />
|25.30%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|867856<br />
|-<br />
|15.26%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|13017856<br />
|-<br />
|13.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|195267856<br />
|-<br />
|7.62%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2929017856<br />
|-<br />
|4.79%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|43935267856<br />
|-<br />
|4.60%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|659029017856<br />
|-<br />
|1.96%<br />
|}<br />
|-<br />
|'''17'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|288<br />
|-<br />
|95.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4641<br />
|-<br />
|34.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|74273<br />
|-<br />
|25.63%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1188385<br />
|-<br />
|11.20%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|19014177<br />
|-<br />
|16.92%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|304226849<br />
|-<br />
|6.07%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|4867629601<br />
|-<br />
|4.54%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|77882073633<br />
|-<br />
|4.33%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1246113178145<br />
|-<br />
|1.22%<br />
|}<br />
|-<br />
|'''18'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|323<br />
|-<br />
|95.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|5526<br />
|-<br />
|29.31%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|93960<br />
|-<br />
|25.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|1597338<br />
|-<br />
|10.75%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|27154764<br />
|-<br />
|14.81%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|461631006<br />
|-<br />
|5.74%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7847727120<br />
|-<br />
|4.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|133411361058<br />
|-<br />
|4.32%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2267993138004<br />
|-<br />
|0.73%<br />
|}<br />
|-<br />
|'''19'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|360<br />
|-<br />
|93.88%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6517<br />
|-<br />
|25.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|117325<br />
|-<br />
|20.43%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2111869<br />
|-<br />
|10.49%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|38013661<br />
|-<br />
|10.58%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|684245917<br />
|-<br />
|5.71%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|12316426525<br />
|-<br />
|4.06%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|221695677469<br />
|-<br />
|3.99%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|3990522194461<br />
|-<br />
|0.45%<br />
|}<br />
|-<br />
|'''20'''<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|399<br />
|-<br />
|95.48%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|7620<br />
|-<br />
|25.69%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|144800<br />
|-<br />
|24.13%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|2751220<br />
|-<br />
|10.24%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|52273200<br />
|-<br />
|17.11%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|993190820<br />
|-<br />
|5.6%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|18870625600<br />
|-<br />
|4.04%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|358541886420<br />
|-<br />
|3.61%<br />
|}<br />
| align="center" |<br />
{| border="2" style="background:#ABCDEF;" <br />
|6812295842000<br />
|-<br />
|1.14%<br />
|}<br />
|}<br />
</center><br />
<br />
<br />
The following table is the key to the colors in the table presented above:<br />
<br />
<center><br />
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"<br />
|'''Color''' || style="text-align: center;" |'''Details'''<br />
|-<br />
|style="background-color: red; text-align: center;" | * || The Moore bound.<br />
|-<br />
|style="background-color: #ABCDEF; text-align: center;" | * || Upper bound introduced by A. Hoffman, R. Singleton, Bannai, E. and Ito, T. <br />
|-<br />
|style="background-color: #999900; text-align: center;" | * || Upper bound introduced by Leif Jorgensen.<br />
|-<br />
|style="background-color: #336666; text-align: center;" | * || Optimal graphs found by Buset and by Molodtsov. <br />
|-<br />
|style="background-color: #ffff00; text-align: center;" | * || Graphs shown optimal.<br />
|-<br />
|}<br />
</center><br />
<br />
==References==<br />
* 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]<br />
* Bannai, E.; Ito, T. (1981), "Regular graphs with excess one", Discrete Mathematics 37:147-158, doi:10.1016/0012-365X(81)90215-6.<br />
* 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.<br />
* 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].<br />
* 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.<br />
* Erd&ouml;s P; Fajtlowicz, S.; Hoffman A. J. (1980), "Maximum degree in graphs of diameter 2", Networks 10: 87-90.<br />
* 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]. <br />
* L. K. Jorgensen (1992), "Diameters of cubic graphs", Discrete Applied Mathematics 37/38: 347-351, doi:10.1016/0166-218X(92)90144-Y.<br />
* 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.<br />
* Kurosawa, K.; Tsujii, S. (1981), "Considerations on diameter of communication networks", Electronics and Communications in Japan 64A (4): 37-45.<br />
* Loz, E.; Širáň, J. (2008), "New record graphs in the degree-diameter problem", Australasian Journal of Combinatorics 41: 63–80.<br />
* 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.<br />
* 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.<br />
* 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.<br />
* 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.<br />
* 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].<br />
* Molodtsov, S. G. (2006), "Largest Graphs of Diameter 2 and Maximum Degree 6", Lecture Notes in Computer Science 4123: 853-857.<br />
* 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.<br />
* Pineda-Villavicencio, G.; Miller, M., "Complete characterization of graphs of maximum degree 3 and defect at most 4", submitted.<br />
* 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. <br />
* 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.<br />
* Brown, W. G. (1966) On graphs that do not contain a Thomsen graph. Canadian Mathematical Bulletin, 9, 281 - 285.<br />
<br />
==External links==<br />
* [http://www-mat.upc.es/grup_de_grafs/ Degree Diameter] online table.<br />
* [http://www.eyal.com.au/wiki/The_Degree/Diameter_Problem Eyal Loz's] Degree-Diameter problem page.<br />
* [http://isu.indstate.edu/ge/DD/index.html Geoffrey Exoo's] Degree-Diameter record graphs page.<br />
<br />
<br />
[[Category:The Degree/Diameter Problem]]</div>
Guillermo
http://combinatoricswiki.org/index.php?title=Main_Page&diff=640
Main Page
2022-02-18T03:59:24Z
<p>Guillermo: </p>
<hr />
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If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.<br />
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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. <br />
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Guillermo
http://combinatoricswiki.org/index.php?title=The_Degree/Diameter_Problem&diff=639
The Degree/Diameter Problem
2022-02-18T03:58:59Z
<p>Guillermo: /* The degree/diameter problem for several classes of graphs (ongoing project): */</p>
<hr />
<div>==Citation==<br />
<br />
If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.<br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
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. <br />
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* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). The degree-diameter problem, Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
==Introduction==<br />
<br />
===Undirected graphs===<br />
<br />
A [http://en.wikipedia.org/wiki/Graph_%28mathematics%29 graph] ''G'' is a collection of points, also called [http://en.wikipedia.org/wiki/Vertex_(graph_theory) vertices] or nodes, and lines connecting these points, also called [http://en.wikipedia.org/wiki/Edge_(graph_theory) edges]. The set of all the vertices of ''G'' is denoted by ''V'', and the set of all edges of ''G'' is denoted by ''E'', and therefore we can write ''G'' as the pair ''(V,E)''. We consider only the case when ''V'' is finite. The set ''E'' can also be regarded as a subset of the 2-element subsets of ''V'', where if ''u'' and ''v'' are two vertices of ''G'', then the edge ''e'' connecting ''u'' to ''v'' is an element of the set ''E'', and therefore we can write ''e'' as ''(u,v)'', and say that ''u'' and ''v'' are adjacent. We can also say that ''u'' and ''v'' are the endvertices of the edge ''e=(u,v)''.<br />
<br />
An edge ''e=(u,v)'' with ''u=v'' is called a loop. Two edges with the same endvertices are called multiple. A simple graph is a graph without loops or multiple edges. Here we only consider simple graphs. <br />
<br />
The number of vertices ''n'' in the graph ''G'' is the order of the graph, and is denoted by ''n=|G|=|V|''.<br />
<br />
The [http://en.wikipedia.org/wiki/degree_(graph_theory) degree] of a vertex, denoted ''d(v)'', is the number of vertices adjacent to ''v''. A graph ''G'' is ''d''-regular if the degree of all the vertices in ''G'' is equal to ''d''.<br />
<br />
A [http://en.wikipedia.org/wiki/Path_(graph_theory) path] ''P'' is a sequence of distinct vertices, such that any consecutive vertices are adjacent, and non-consecutive vertices are not. A [http://en.wikipedia.org/wiki/Cycle_(graph_theory) cycle] is a path of at least three vertices starting and ending in the same vertex. In a graph ''G'' the [http://en.wikipedia.org/wiki/distance_(graph_theory) distance] between two vertices ''u'' and ''v'' is the number of edges in a shortest path starting at ''u'' and terminating at ''v''. We denote this distance by ''dist(u,v)''. If no such path exists, and hence, the graph is disconnected, we say that the distance is <math>\infty</math>. The [http://en.wikipedia.org/wiki/distance_(graph_theory) diameter] ''k'' of ''G'' is the maximum pairwise distance between the vertices of the graph.<br />
<br />
The '''degree diameter problem''' is the problem of finding the largest possible number ''N(d,k)'' of vertices in a graph of maximum degree ''d'' and diameter ''k''.<br />
<br />
We call a graph of maximum degree ''d'' and diameter ''k'' a ''(d,k)''-graph.<br />
<br />
An upper bound on the order of a ''(d,k)''-graph is given by the expression ''(d(d-1)<sup>k</sup>-2)(d-2)<sup>-1</sup>'', known as the [http://en.wikipedia.org/wiki/Moore_graph Moore bound], and denoted by ''M(d,k)''.<br />
<br />
Graphs whose order attains the Moore bound are called [http://en.wikipedia.org/wiki/Moore_graph Moore graphs]. Moore graphs proved to be very rare. They are the complete graphs on ''d+1'' vertices, the cycles on ''2k+1'' vertices, and for diameter 2, the [http://en.wikipedia.org/wiki/Petersen_graph Petersen graph], the [http://en.wikipedia.org/wiki/Hoffman-Singleton_graph Hoffman-Singleton graph] and probably a graph of degree ''d''&nbsp;=&nbsp;57. For 2&nbsp;<&nbsp;''k'' and 2&nbsp;<&nbsp;''d'', there are no Moore graphs. <br />
<br />
Consequently, research activities in the degree/diameter problem cover (i) ''Constructions of ever larger graphs, which provide better lower bounds on ''N(d,k)'' '' and (ii) ''Proofs of the non-existence or otherwise of graphs whose order misses the Moore bound by a small number of vertices''. A comprehensive information source, covering the current state of knowledge in the degree/diameter problem for general graphs [[The Degree Diameter Problem for General Graphs|can be found here]].<br />
<br />
===Directed graphs===<br />
<br />
If we give an orientation to each edge of ''G'', then, we call ''G'' a directed graph, or simply digraph, and accordingly, the edges of ''G'' directed edges, or simply arcs. The number of arcs starting at a vertex ''u'' represents the out-degree of ''u'', while the number of arcs arriving at a vertex ''u'' gives the in-degree of ''u''. Two arcs ''(u,v)'' and ''(w,t)'' are adjacent if, and only if, v=w or ''u=t''. A directed path is a sequence of adjacent arcs, with all the vertices distinct. The notions of a cycle, distance and diameter are defined as in the case of undirected graphs but considering the orientation of the arcs.<br />
<br />
In the context of digraphs the degree/diameter problem can be considered as the problem of finding the largest possible number ''DN(d,k)'' of vertices in a digraph of maximum out-degree ''d'' and diameter ''k''.<br />
<br />
We call a digraph of maximum out-degree ''d'' and diameter ''k'' a ''(d,k)''-digraph.<br />
<br />
An upper bound on the order of a ''(d,k)''-digraph is given by <br />
the expression ''(d<sup>k+1</sup>-1)(d-1)<sup>-1</sup>'', known as the directed Moore bound, and denoted by ''DM(d,k)''. <br />
<br />
Digraphs attaining the directed Moore bound are called Moore digraphs, and they are even rarer than the Moore graphs. Moore digraphs are the directed cycles on ''k+1'' vertices and the complete digraphs on ''d+1'' vertices. For 1&nbsp;<&nbsp;''k'' and 1&nbsp;<&nbsp;''d'', there are no Moore digraphs. A comprehensive information source, covering the current state of knowledge in the degree/diameter problem for general digraphs [[The Degree Diameter Problem for General Digraphs|can be found here]].<br />
<br />
===Mixed graphs===<br />
<br />
A mixed graph (also known as partially directed graph) <math>G</math> may contain (undirected) edges as well as directed edges (also known as arcs or darts). Hence, mixed graphs generalize both undirected and directed graphs. Nevertheless, here we suppose that mixed graphs contain at least one edge and one arc. The undirected degree of a vertex <math>v</math>, denoted by <math>d(v)</math> is the number of edges incident to <math>v</math>. The out-degree [resp. in-degree] of <math>v</math>, denoted by <math>d^+(v)</math> [resp. <math>d^-(v)</math>], is the number of arcs emanating from [resp. to] <math>v</math>. If <math>d^+(v)=d^-(v)=z</math> and <math>d(v)=r</math>, for all vertex <math>v</math> in <math>G</math>, then <math>G</math> is said to be totally regular of degree <math>d</math>, where <math>d=r+z</math>. A em trail from <math>u</math> to <math>v</math> of length <math>l</math> is a sequence of <math>l+1</math> vertices, such that the first vertex of the sequence is <math>u</math> and the last one is <math>v</math> and each pair of consecutive vertices of the sequence is either an edge or an arc of <math>G</math>. The notions of path, distance and diameter are defined as in the previous cases, but allowing edges and arcs both together. Note that the distance from <math>u</math> to <math>v</math> may be different than the reverse distance from <math>v</math> to <math>u</math> when shortest paths between these vertices involve arcs.<br />
<br />
In the context of mixed graphs the degree/diameter problem can be considered as the problem of finding the largest possible number <math>N(r,z,k)</math> of vertices in a mixed graph of maximum undirected degree <math>r</math>, maximum directed out-degree <math>z</math> and diameter <math>k</math>.<br />
<br />
An upper bound for <math>N(r,z,k)</math> is given by <br />
the expression <math>A(u^{k+1}-1)(u-1)^{-1}+B(w^{k+1}-1)(w-1)^{-1} </math>, where <math>v=(z+r)^2+2(z-r)+1</math>, <math>u=(1/2)(z+r-1-v^{1/2})</math>, <math>w=(1/2)(z+r-1+v^{1/2})</math>, <math>A=(v^{1/2}-(z+r+1))(2v^{1/2})^{-1}</math> and <br />
<math>B=(v^{1/2}+(z+r+1))(2v^{1/2})^{-1}</math>. This is known as the mixed Moore bound, and it is denoted by <math>M(r,z,k)</math>.<br />
<br />
Mixed graphs of order <math>M(r,z,k)</math> with maximum undirected degree <math>r</math>, maximum directed out-degree <math>z</math> and diameter <math>k</math> are called mixed Moore graphs. Such extremal mixed graphs are totally regular of degree <math>d=r+z</math> and they have the property that for any ordered pair <math>(u,v)</math> of vertices there is a unique trail from <math>u</math> to <math>v</math> of length at most the (finite) diameter <math>k</math>. These extremal mixed graphs may only exist for diameter two and some (infinitely many) values of <math>M(r,z,2)</math>, although their existence has been proved only for 'few' cases. A comprehensive information source, covering the current state of knowledge in the degree/diameter problem for general mixed graphs [[The Degree Diameter Problem for General Mixed Graphs|can be found here]].<br />
<br />
==The degree/diameter problem for several classes of graphs (ongoing project):==<br />
The following database of tables and pages is maintained and moderated by '''[[Eyal Loz]]''', '''[[Hebert Pérez-Rosés]]''', '''[[Guillermo Pineda-Villavicencio]]''', and '''[[Nacho López]]''' as part of the project '''[[The Degree/Diameter Problem for Several Classes of Graphs Project|The degree/diameter problem for several classes of graphs]]'''. <br />
<br />
If, in addition to the limits on the degree and the diameter, we add further constraints to the graphs in question, we can state the degree/diameter problem for several classes of graphs, such as bipartite graphs, vertex-transitive graphs, Cayley graphs, arc-transitive graphs, and the corresponding versions for digraphs. In these cases we denote the largest possible number of nodes by ''N<sup>b</sup>(d,k)'' (for bipartite graphs), ''N<sup>vt</sup>(d,k)'' (for vertex-transitive graphs), ''N<sup>c</sup>(d,k)'' (for Cayley graphs), and ''N<sup>at</sup>(d,k)'' (for arc-transitive graphs), respectively.<br />
<br />
For only a few classes of graphs, better general upper bounds are known. For information on the best upper bounds known at present for a certain class, follow the corresponding link below.<br />
<br />
<br />
'''The undirected case:'''<br />
<br />
* [[The Degree Diameter Problem for General Graphs|The degree/diameter problem for general graphs]].<br />
<br />
* [[The Degree Diameter Problem for Cayley Graphs|The degree/diameter problem for Cayley graphs]].<br />
<br />
* [[The Degree Diameter Problem for Circulant Graphs|The degree/diameter problem for circulant graphs]].<br />
<br />
* [[The Degree Diameter Problem for Bipartite Graphs|The degree/diameter problem for bipartite graphs]].<br />
<br />
* [[The Degree Diameter Problem for Vertex Transitive Graphs|The degree/diameter problem for vertex-transitive graphs]].<br />
<br />
* [[The Degree Diameter Problem for Arc Transitive Graphs|The degree/diameter problem for arc-transitive graphs]].<br />
<br />
* [[The Degree Diameter Problem for Planar Graphs|The degree/diameter problem for planar graphs]].<br />
<br />
* [[The Degree Diameter Problem for Toroidal Graphs|The degree/diameter problem for toroidal graphs]]. <br />
<br />
'''The directed case:'''<br />
<br />
* [[The Degree Diameter Problem for General Digraphs|The degree/diameter problem for general digraphs]].<br />
<br />
* [[The Degree Diameter Problem for Vertex Symmetric Digraphs|The degree/diameter problem for vertex-symmetric digraphs]].<br />
<br />
* [[The Degree Diameter Problem for Circulant Digraphs|The degree/diameter problem for circulant digraphs]].<br />
<br />
'''The mixed case:'''<br />
<br />
* [[The Degree Diameter Problem for General Mixed Graphs|The degree/diameter problem for general mixed graphs]].<br />
<br />
* [[The Degree Diameter Problem for Mixed Circulant|The degree/diameter problem for mixed circulant graphs]].<br />
<br />
== References ==<br />
* Bannai, E.; Ito, T. (1973), "On finite Moore graphs", J. Fac. Sci. Univ. Tokyo Ser. A 20: 191–208, MR0323615<br />
* Bosàk, J., (1978), "Partially directed Moore graphs", Math. Slovaca, (29):181–196. [http://dml.cz/bitstream/handle/10338.dmlcz/129861/MathSlov_29-1979-2_10.pdf PDF version]<br />
* Bridges, W. G.; Toueg, S. (1980), "On the impossibility of directed Moore graphs", Journal of Combinatorial Theory, Series B 29 (3), 339--341, doi:10.1016/0095-8956(80)90091-X.<br />
* Hoffman, Alan J.; Singleton, Robert 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] <br />
* Singleton, Robert R. (1968), "There is no irregular Moore graph", American Mathematical Monthly 75 (1): 42–43, MR0225679<br />
* Miller, Mirka; Širáň, Jozef (2005), "Moore graphs and beyond: A survey of the degree/diameter problem", Electronic Journal of Combinatorics Dynamic survey DS14 [http://www.combinatorics.org/Surveys/ds14.pdf PDF version].<br />
<br />
<br />
<br />
[[Category:The Degree/Diameter Problem]]</div>
Guillermo
http://combinatoricswiki.org/index.php?title=The_Degree/Diameter_Problem&diff=638
The Degree/Diameter Problem
2022-02-18T03:58:15Z
<p>Guillermo: </p>
<hr />
<div>==Citation==<br />
<br />
If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.<br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
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. <br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). The degree-diameter problem, Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
==Introduction==<br />
<br />
===Undirected graphs===<br />
<br />
A [http://en.wikipedia.org/wiki/Graph_%28mathematics%29 graph] ''G'' is a collection of points, also called [http://en.wikipedia.org/wiki/Vertex_(graph_theory) vertices] or nodes, and lines connecting these points, also called [http://en.wikipedia.org/wiki/Edge_(graph_theory) edges]. The set of all the vertices of ''G'' is denoted by ''V'', and the set of all edges of ''G'' is denoted by ''E'', and therefore we can write ''G'' as the pair ''(V,E)''. We consider only the case when ''V'' is finite. The set ''E'' can also be regarded as a subset of the 2-element subsets of ''V'', where if ''u'' and ''v'' are two vertices of ''G'', then the edge ''e'' connecting ''u'' to ''v'' is an element of the set ''E'', and therefore we can write ''e'' as ''(u,v)'', and say that ''u'' and ''v'' are adjacent. We can also say that ''u'' and ''v'' are the endvertices of the edge ''e=(u,v)''.<br />
<br />
An edge ''e=(u,v)'' with ''u=v'' is called a loop. Two edges with the same endvertices are called multiple. A simple graph is a graph without loops or multiple edges. Here we only consider simple graphs. <br />
<br />
The number of vertices ''n'' in the graph ''G'' is the order of the graph, and is denoted by ''n=|G|=|V|''.<br />
<br />
The [http://en.wikipedia.org/wiki/degree_(graph_theory) degree] of a vertex, denoted ''d(v)'', is the number of vertices adjacent to ''v''. A graph ''G'' is ''d''-regular if the degree of all the vertices in ''G'' is equal to ''d''.<br />
<br />
A [http://en.wikipedia.org/wiki/Path_(graph_theory) path] ''P'' is a sequence of distinct vertices, such that any consecutive vertices are adjacent, and non-consecutive vertices are not. A [http://en.wikipedia.org/wiki/Cycle_(graph_theory) cycle] is a path of at least three vertices starting and ending in the same vertex. In a graph ''G'' the [http://en.wikipedia.org/wiki/distance_(graph_theory) distance] between two vertices ''u'' and ''v'' is the number of edges in a shortest path starting at ''u'' and terminating at ''v''. We denote this distance by ''dist(u,v)''. If no such path exists, and hence, the graph is disconnected, we say that the distance is <math>\infty</math>. The [http://en.wikipedia.org/wiki/distance_(graph_theory) diameter] ''k'' of ''G'' is the maximum pairwise distance between the vertices of the graph.<br />
<br />
The '''degree diameter problem''' is the problem of finding the largest possible number ''N(d,k)'' of vertices in a graph of maximum degree ''d'' and diameter ''k''.<br />
<br />
We call a graph of maximum degree ''d'' and diameter ''k'' a ''(d,k)''-graph.<br />
<br />
An upper bound on the order of a ''(d,k)''-graph is given by the expression ''(d(d-1)<sup>k</sup>-2)(d-2)<sup>-1</sup>'', known as the [http://en.wikipedia.org/wiki/Moore_graph Moore bound], and denoted by ''M(d,k)''.<br />
<br />
Graphs whose order attains the Moore bound are called [http://en.wikipedia.org/wiki/Moore_graph Moore graphs]. Moore graphs proved to be very rare. They are the complete graphs on ''d+1'' vertices, the cycles on ''2k+1'' vertices, and for diameter 2, the [http://en.wikipedia.org/wiki/Petersen_graph Petersen graph], the [http://en.wikipedia.org/wiki/Hoffman-Singleton_graph Hoffman-Singleton graph] and probably a graph of degree ''d''&nbsp;=&nbsp;57. For 2&nbsp;<&nbsp;''k'' and 2&nbsp;<&nbsp;''d'', there are no Moore graphs. <br />
<br />
Consequently, research activities in the degree/diameter problem cover (i) ''Constructions of ever larger graphs, which provide better lower bounds on ''N(d,k)'' '' and (ii) ''Proofs of the non-existence or otherwise of graphs whose order misses the Moore bound by a small number of vertices''. A comprehensive information source, covering the current state of knowledge in the degree/diameter problem for general graphs [[The Degree Diameter Problem for General Graphs|can be found here]].<br />
<br />
===Directed graphs===<br />
<br />
If we give an orientation to each edge of ''G'', then, we call ''G'' a directed graph, or simply digraph, and accordingly, the edges of ''G'' directed edges, or simply arcs. The number of arcs starting at a vertex ''u'' represents the out-degree of ''u'', while the number of arcs arriving at a vertex ''u'' gives the in-degree of ''u''. Two arcs ''(u,v)'' and ''(w,t)'' are adjacent if, and only if, v=w or ''u=t''. A directed path is a sequence of adjacent arcs, with all the vertices distinct. The notions of a cycle, distance and diameter are defined as in the case of undirected graphs but considering the orientation of the arcs.<br />
<br />
In the context of digraphs the degree/diameter problem can be considered as the problem of finding the largest possible number ''DN(d,k)'' of vertices in a digraph of maximum out-degree ''d'' and diameter ''k''.<br />
<br />
We call a digraph of maximum out-degree ''d'' and diameter ''k'' a ''(d,k)''-digraph.<br />
<br />
An upper bound on the order of a ''(d,k)''-digraph is given by <br />
the expression ''(d<sup>k+1</sup>-1)(d-1)<sup>-1</sup>'', known as the directed Moore bound, and denoted by ''DM(d,k)''. <br />
<br />
Digraphs attaining the directed Moore bound are called Moore digraphs, and they are even rarer than the Moore graphs. Moore digraphs are the directed cycles on ''k+1'' vertices and the complete digraphs on ''d+1'' vertices. For 1&nbsp;<&nbsp;''k'' and 1&nbsp;<&nbsp;''d'', there are no Moore digraphs. A comprehensive information source, covering the current state of knowledge in the degree/diameter problem for general digraphs [[The Degree Diameter Problem for General Digraphs|can be found here]].<br />
<br />
===Mixed graphs===<br />
<br />
A mixed graph (also known as partially directed graph) <math>G</math> may contain (undirected) edges as well as directed edges (also known as arcs or darts). Hence, mixed graphs generalize both undirected and directed graphs. Nevertheless, here we suppose that mixed graphs contain at least one edge and one arc. The undirected degree of a vertex <math>v</math>, denoted by <math>d(v)</math> is the number of edges incident to <math>v</math>. The out-degree [resp. in-degree] of <math>v</math>, denoted by <math>d^+(v)</math> [resp. <math>d^-(v)</math>], is the number of arcs emanating from [resp. to] <math>v</math>. If <math>d^+(v)=d^-(v)=z</math> and <math>d(v)=r</math>, for all vertex <math>v</math> in <math>G</math>, then <math>G</math> is said to be totally regular of degree <math>d</math>, where <math>d=r+z</math>. A em trail from <math>u</math> to <math>v</math> of length <math>l</math> is a sequence of <math>l+1</math> vertices, such that the first vertex of the sequence is <math>u</math> and the last one is <math>v</math> and each pair of consecutive vertices of the sequence is either an edge or an arc of <math>G</math>. The notions of path, distance and diameter are defined as in the previous cases, but allowing edges and arcs both together. Note that the distance from <math>u</math> to <math>v</math> may be different than the reverse distance from <math>v</math> to <math>u</math> when shortest paths between these vertices involve arcs.<br />
<br />
In the context of mixed graphs the degree/diameter problem can be considered as the problem of finding the largest possible number <math>N(r,z,k)</math> of vertices in a mixed graph of maximum undirected degree <math>r</math>, maximum directed out-degree <math>z</math> and diameter <math>k</math>.<br />
<br />
An upper bound for <math>N(r,z,k)</math> is given by <br />
the expression <math>A(u^{k+1}-1)(u-1)^{-1}+B(w^{k+1}-1)(w-1)^{-1} </math>, where <math>v=(z+r)^2+2(z-r)+1</math>, <math>u=(1/2)(z+r-1-v^{1/2})</math>, <math>w=(1/2)(z+r-1+v^{1/2})</math>, <math>A=(v^{1/2}-(z+r+1))(2v^{1/2})^{-1}</math> and <br />
<math>B=(v^{1/2}+(z+r+1))(2v^{1/2})^{-1}</math>. This is known as the mixed Moore bound, and it is denoted by <math>M(r,z,k)</math>.<br />
<br />
Mixed graphs of order <math>M(r,z,k)</math> with maximum undirected degree <math>r</math>, maximum directed out-degree <math>z</math> and diameter <math>k</math> are called mixed Moore graphs. Such extremal mixed graphs are totally regular of degree <math>d=r+z</math> and they have the property that for any ordered pair <math>(u,v)</math> of vertices there is a unique trail from <math>u</math> to <math>v</math> of length at most the (finite) diameter <math>k</math>. These extremal mixed graphs may only exist for diameter two and some (infinitely many) values of <math>M(r,z,2)</math>, although their existence has been proved only for 'few' cases. A comprehensive information source, covering the current state of knowledge in the degree/diameter problem for general mixed graphs [[The Degree Diameter Problem for General Mixed Graphs|can be found here]].<br />
<br />
==The degree/diameter problem for several classes of graphs (ongoing project):==<br />
The following database of tables and pages is maintained and moderated by '''[[Eyal Loz]]''', '''[[Hebert Pérez-Rosés]]''', '''[[Guillermo Pineda-Villavicencio]]''', '''[[Ramiro Feria-Puron]]''' and '''[[Nacho López]]''' as part of the project '''[[The Degree/Diameter Problem for Several Classes of Graphs Project|The degree/diameter problem for several classes of graphs]]'''. <br />
<br />
If, in addition to the limits on the degree and the diameter, we add further constraints to the graphs in question, we can state the degree/diameter problem for several classes of graphs, such as bipartite graphs, vertex-transitive graphs, Cayley graphs, arc-transitive graphs, and the corresponding versions for digraphs. In these cases we denote the largest possible number of nodes by ''N<sup>b</sup>(d,k)'' (for bipartite graphs), ''N<sup>vt</sup>(d,k)'' (for vertex-transitive graphs), ''N<sup>c</sup>(d,k)'' (for Cayley graphs), and ''N<sup>at</sup>(d,k)'' (for arc-transitive graphs), respectively.<br />
<br />
For only a few classes of graphs, better general upper bounds are known. For information on the best upper bounds known at present for a certain class, follow the corresponding link below.<br />
<br />
<br />
'''The undirected case:'''<br />
<br />
* [[The Degree Diameter Problem for General Graphs|The degree/diameter problem for general graphs]].<br />
<br />
* [[The Degree Diameter Problem for Cayley Graphs|The degree/diameter problem for Cayley graphs]].<br />
<br />
* [[The Degree Diameter Problem for Circulant Graphs|The degree/diameter problem for circulant graphs]].<br />
<br />
* [[The Degree Diameter Problem for Bipartite Graphs|The degree/diameter problem for bipartite graphs]].<br />
<br />
* [[The Degree Diameter Problem for Vertex Transitive Graphs|The degree/diameter problem for vertex-transitive graphs]].<br />
<br />
* [[The Degree Diameter Problem for Arc Transitive Graphs|The degree/diameter problem for arc-transitive graphs]].<br />
<br />
* [[The Degree Diameter Problem for Planar Graphs|The degree/diameter problem for planar graphs]].<br />
<br />
* [[The Degree Diameter Problem for Toroidal Graphs|The degree/diameter problem for toroidal graphs]]. <br />
<br />
'''The directed case:'''<br />
<br />
* [[The Degree Diameter Problem for General Digraphs|The degree/diameter problem for general digraphs]].<br />
<br />
* [[The Degree Diameter Problem for Vertex Symmetric Digraphs|The degree/diameter problem for vertex-symmetric digraphs]].<br />
<br />
* [[The Degree Diameter Problem for Circulant Digraphs|The degree/diameter problem for circulant digraphs]].<br />
<br />
'''The mixed case:'''<br />
<br />
* [[The Degree Diameter Problem for General Mixed Graphs|The degree/diameter problem for general mixed graphs]].<br />
<br />
* [[The Degree Diameter Problem for Mixed Circulant|The degree/diameter problem for mixed circulant graphs]].<br />
<br />
== References ==<br />
* Bannai, E.; Ito, T. (1973), "On finite Moore graphs", J. Fac. Sci. Univ. Tokyo Ser. A 20: 191–208, MR0323615<br />
* Bosàk, J., (1978), "Partially directed Moore graphs", Math. Slovaca, (29):181–196. [http://dml.cz/bitstream/handle/10338.dmlcz/129861/MathSlov_29-1979-2_10.pdf PDF version]<br />
* Bridges, W. G.; Toueg, S. (1980), "On the impossibility of directed Moore graphs", Journal of Combinatorial Theory, Series B 29 (3), 339--341, doi:10.1016/0095-8956(80)90091-X.<br />
* Hoffman, Alan J.; Singleton, Robert 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] <br />
* Singleton, Robert R. (1968), "There is no irregular Moore graph", American Mathematical Monthly 75 (1): 42–43, MR0225679<br />
* Miller, Mirka; Širáň, Jozef (2005), "Moore graphs and beyond: A survey of the degree/diameter problem", Electronic Journal of Combinatorics Dynamic survey DS14 [http://www.combinatorics.org/Surveys/ds14.pdf PDF version].<br />
<br />
<br />
<br />
[[Category:The Degree/Diameter Problem]]</div>
Guillermo
http://combinatoricswiki.org/index.php?title=Main_Page&diff=637
Main Page
2022-02-18T03:57:26Z
<p>Guillermo: </p>
<hr />
<div>==About Combinatorics Wiki==<br />
<br />
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:<br />
<br />
* 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.<br />
<br />
* 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.<br />
<br />
* Keeping a detailed history of previous work, findings, publications and results, in a simple user friendly wiki format.<br />
<br />
* Allowing registered users to review and comment on unpublished and published work by other users.<br />
<br />
* Giving supervisors and students ideas for new projects and open problems.<br />
<br />
* Creating a stable community of researchers in different areas, and promoting collaborations.<br />
<br />
===Citation=== <br />
<br />
If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.<br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
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. <br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). The degree-diameter problem, Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
==List of problem areas==<br />
<br />
* [[Enumeration of latin squares and rectangles]]<br />
<br />
* [[The cage problem|The cage Problem or The degree/girth problem]]<br />
<br />
* [[The Degree/Diameter Problem]]<br />
<br />
* [[The maximum degree-and-diameter-bounded subgraph problem]] <br />
<br />
* [[Extremal C_t-free graphs]]<br />
<br />
==List of video channels==<br />
<br />
* [[Lectures Available Online|Lectures available online]]<br />
<br />
* [[Documentaries Available Online|Documentaries available online]]<br />
<br />
<br />
==Supporting organizations==<br />
<br />
* [http://combinatorics-australasia.org/ CMSA - Combinatorial Mathematics Society of Australasia]<br />
<br />
* [http://graphtheorygroup.com GTA - Graph Theory and Applications - The University of Newcastle, Australia]<br />
<br />
* [http://www.indstate.edu/home.htm Indiana State University]<br />
<br />
* [http://www-mat.upc.es/grup_de_grafs Research Group on Graph Theory and Combinatorics - UPC, Spain]<br />
<br />
==Newsletters==<br />
<br />
[[Combinatorial_Mathematics_Society_of_Australasia_Newsletters|Newsletters of the Combinatorial Mathematics Society of Australasia]]<br />
<br />
==Meetings, seminars and talks==<br />
<br />
Links to upcoming talks which may be of interest to researchers in the problem areas covered by Combinatorics Wiki.<br />
<br />
[[Mirka_Miller%27s_Combinatorics_Webinar_Series|Mirka Miller's Combinatorics Webinar Series]]<br />
<br />
==Employment Opportunities== <br />
<br />
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.<br />
<br />
<br />
==Combinatorics Wiki rules==<br />
<br />
Please read our rules of [[Rules and Regulations|use of Combinatorics Wiki]]. <br />
<br />
<br />
==Note for potential new contributors and moderators==<br />
<br />
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).</div>
Guillermo
http://combinatoricswiki.org/index.php?title=Main_Page&diff=636
Main Page
2022-02-18T03:56:24Z
<p>Guillermo: </p>
<hr />
<div>==About Combinatorics Wiki==<br />
<br />
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:<br />
<br />
* 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.<br />
<br />
* 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.<br />
<br />
* Keeping a detailed history of previous work, findings, publications and results, in a simple user friendly wiki format.<br />
<br />
* Allowing registered users to review and comment on unpublished and published work by other users.<br />
<br />
* Giving supervisors and students ideas for new projects and open problems.<br />
<br />
* Creating a stable community of researchers in different areas, and promoting collaborations.<br />
<br />
===Citation=== <br />
<br />
If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.<br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
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. <br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). The degree-diameter problem, Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
==List of problem areas==<br />
<br />
* [[Enumeration of latin squares and rectangles]]<br />
<br />
* [[The cage problem|The cage Problem or The degree/girth problem]]<br />
<br />
* [[The degree-diameter Problem]]<br />
<br />
* [[The maximum degree-and-diameter-bounded subgraph problem]] <br />
<br />
* [[Extremal C_t-free graphs]]<br />
<br />
==List of video channels==<br />
<br />
* [[Lectures Available Online|Lectures available online]]<br />
<br />
* [[Documentaries Available Online|Documentaries available online]]<br />
<br />
<br />
==Supporting organizations==<br />
<br />
* [http://combinatorics-australasia.org/ CMSA - Combinatorial Mathematics Society of Australasia]<br />
<br />
* [http://graphtheorygroup.com GTA - Graph Theory and Applications - The University of Newcastle, Australia]<br />
<br />
* [http://www.indstate.edu/home.htm Indiana State University]<br />
<br />
* [http://www-mat.upc.es/grup_de_grafs Research Group on Graph Theory and Combinatorics - UPC, Spain]<br />
<br />
==Newsletters==<br />
<br />
[[Combinatorial_Mathematics_Society_of_Australasia_Newsletters|Newsletters of the Combinatorial Mathematics Society of Australasia]]<br />
<br />
==Meetings, seminars and talks==<br />
<br />
Links to upcoming talks which may be of interest to researchers in the problem areas covered by Combinatorics Wiki.<br />
<br />
[[Mirka_Miller%27s_Combinatorics_Webinar_Series|Mirka Miller's Combinatorics Webinar Series]]<br />
<br />
==Employment Opportunities== <br />
<br />
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.<br />
<br />
<br />
==Combinatorics Wiki rules==<br />
<br />
Please read our rules of [[Rules and Regulations|use of Combinatorics Wiki]]. <br />
<br />
<br />
==Note for potential new contributors and moderators==<br />
<br />
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).</div>
Guillermo
http://combinatoricswiki.org/index.php?title=Main_Page&diff=635
Main Page
2022-02-18T03:55:10Z
<p>Guillermo: </p>
<hr />
<div>==About Combinatorics Wiki==<br />
<br />
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:<br />
<br />
* 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.<br />
<br />
* 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.<br />
<br />
* Keeping a detailed history of previous work, findings, publications and results, in a simple user friendly wiki format.<br />
<br />
* Allowing registered users to review and comment on unpublished and published work by other users.<br />
<br />
* Giving supervisors and students ideas for new projects and open problems.<br />
<br />
* Creating a stable community of researchers in different areas, and promoting collaborations.<br />
<br />
===Citation=== <br />
<br />
If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.<br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
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. <br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio (2010). The degree-diameter problem, Combinatorics Wiki, http://combinatoricswiki.org.<br />
<br />
==List of problem areas==<br />
<br />
* [[Enumeration of latin squares and rectangles]]<br />
<br />
* [[The cage problem|The cage Problem or The degree/girth problem]]<br />
<br />
* [[The degree-diameter Problem]]<br />
<br />
* [[The maximum degree-and-diameter-bounded subgraph problem]] <br />
<br />
* [[Minor-closed classes of Matroids]]<br />
<br />
* [[Ramsey Theory]]<br />
<br />
* [[Extremal C_t-free graphs]]<br />
<br />
==List of video channels==<br />
<br />
* [[Lectures Available Online|Lectures available online]]<br />
<br />
* [[Documentaries Available Online|Documentaries available online]]<br />
<br />
<br />
==Supporting organizations==<br />
<br />
* [http://combinatorics-australasia.org/ CMSA - Combinatorial Mathematics Society of Australasia]<br />
<br />
* [http://graphtheorygroup.com GTA - Graph Theory and Applications - The University of Newcastle, Australia]<br />
<br />
* [http://www.indstate.edu/home.htm Indiana State University]<br />
<br />
* [http://www-mat.upc.es/grup_de_grafs Research Group on Graph Theory and Combinatorics - UPC, Spain]<br />
<br />
==Newsletters==<br />
<br />
[[Combinatorial_Mathematics_Society_of_Australasia_Newsletters|Newsletters of the Combinatorial Mathematics Society of Australasia]]<br />
<br />
==Meetings, seminars and talks==<br />
<br />
Links to upcoming talks which may be of interest to researchers in the problem areas covered by Combinatorics Wiki.<br />
<br />
[[Mirka_Miller%27s_Combinatorics_Webinar_Series|Mirka Miller's Combinatorics Webinar Series]]<br />
<br />
==Employment Opportunities== <br />
<br />
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.<br />
<br />
<br />
==Combinatorics Wiki rules==<br />
<br />
Please read our rules of [[Rules and Regulations|use of Combinatorics Wiki]]. <br />
<br />
<br />
==Note for potential new contributors and moderators==<br />
<br />
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).</div>
Guillermo
http://combinatoricswiki.org/index.php?title=Main_Page&diff=634
Main Page
2022-02-18T03:53:36Z
<p>Guillermo: </p>
<hr />
<div>==About Combinatorics Wiki==<br />
<br />
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:<br />
<br />
* 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.<br />
<br />
* 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.<br />
<br />
* Keeping a detailed history of previous work, findings, publications and results, in a simple user friendly wiki format.<br />
<br />
* Allowing registered users to review and comment on unpublished and published work by other users.<br />
<br />
* Giving supervisors and students ideas for new projects and open problems.<br />
<br />
* Creating a stable community of researchers in different areas, and promoting collaborations.<br />
<br />
===Citation=== <br />
<br />
If you are using combinatoricsWiki, then we would like to ask you to cite the site as follows.<br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio. {Combinatorics Wiki}, \url{http://combinatoricswiki.org}, 2010.<br />
<br />
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. <br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio. The degree-diameter problem, {Combinatorics Wiki}, \url{http://combinatoricswiki.org}, 2010.<br />
<br />
==List of problem areas==<br />
<br />
* [[Enumeration of latin squares and rectangles]]<br />
<br />
* [[The cage problem|The cage Problem or The degree/girth problem]]<br />
<br />
* [[The degree-diameter Problem]]<br />
<br />
* [[The maximum degree-and-diameter-bounded subgraph problem]] <br />
<br />
* [[Minor-closed classes of Matroids]]<br />
<br />
* [[Ramsey Theory]]<br />
<br />
* [[Extremal C_t-free graphs]]<br />
<br />
==List of video channels==<br />
<br />
* [[Lectures Available Online|Lectures available online]]<br />
<br />
* [[Documentaries Available Online|Documentaries available online]]<br />
<br />
<br />
==Supporting organizations==<br />
<br />
* [http://combinatorics-australasia.org/ CMSA - Combinatorial Mathematics Society of Australasia]<br />
<br />
* [http://graphtheorygroup.com GTA - Graph Theory and Applications - The University of Newcastle, Australia]<br />
<br />
* [http://www.indstate.edu/home.htm Indiana State University]<br />
<br />
* [http://www-mat.upc.es/grup_de_grafs Research Group on Graph Theory and Combinatorics - UPC, Spain]<br />
<br />
==Newsletters==<br />
<br />
[[Combinatorial_Mathematics_Society_of_Australasia_Newsletters|Newsletters of the Combinatorial Mathematics Society of Australasia]]<br />
<br />
==Meetings, seminars and talks==<br />
<br />
Links to upcoming talks which may be of interest to researchers in the problem areas covered by Combinatorics Wiki.<br />
<br />
[[Mirka_Miller%27s_Combinatorics_Webinar_Series|Mirka Miller's Combinatorics Webinar Series]]<br />
<br />
==Employment Opportunities== <br />
<br />
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.<br />
<br />
<br />
==Combinatorics Wiki rules==<br />
<br />
Please read our rules of [[Rules and Regulations|use of Combinatorics Wiki]]. <br />
<br />
<br />
==Note for potential new contributors and moderators==<br />
<br />
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).</div>
Guillermo
http://combinatoricswiki.org/index.php?title=Main_Page&diff=633
Main Page
2022-02-18T03:51:22Z
<p>Guillermo: </p>
<hr />
<div>==About Combinatorics Wiki==<br />
<br />
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:<br />
<br />
* 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.<br />
<br />
* 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.<br />
<br />
* Keeping a detailed history of previous work, findings, publications and results, in a simple user friendly wiki format.<br />
<br />
* Allowing registered users to review and comment on unpublished and published work by other users.<br />
<br />
* Giving supervisors and students ideas for new projects and open problems.<br />
<br />
* Creating a stable community of researchers in different areas, and promoting collaborations.<br />
<br />
==Citation <br />
<br />
<br />
If you are using combinatoricsWiki we would like to ask you to cite the site as follows.<br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio. {Combinatorics Wiki}, \url{http://combinatoricswiki.org}, 2010.<br />
<br />
If you are using a specific page, say the "The degree-diameter problem", then it would be better to cite the page as follows. <br />
<br />
* E. Loz, H. P\'erez-Ros\'es and G.Pineda-Villavicencio. The degree-diameter problem, {Combinatorics Wiki}, \url{http://combinatoricswiki.org}, 2010.<br />
<br />
==List of problem areas==<br />
<br />
* [[Enumeration of Latin Squares and Rectangles]]<br />
<br />
* [[The Cage Problem|The Cage Problem or The Degree/Girth Problem]]<br />
<br />
* [[The Degree/Diameter Problem]]<br />
<br />
* [[The Maximum Degree-and-Diameter-Bounded Subgraph Problem]] <br />
<br />
* [[Minor-Closed Classes of Matroids]]<br />
<br />
* [[Ramsey Theory]]<br />
<br />
* [[Extremal C_t-free graphs]]<br />
<br />
==List of video channels==<br />
<br />
* [[Lectures Available Online|Lectures available online]]<br />
<br />
* [[Documentaries Available Online|Documentaries available online]]<br />
<br />
<br />
==Supporting organizations==<br />
<br />
* [http://combinatorics-australasia.org/ CMSA - Combinatorial Mathematics Society of Australasia]<br />
<br />
* [http://graphtheorygroup.com GTA - Graph Theory and Applications - The University of Newcastle, Australia]<br />
<br />
* [http://www.indstate.edu/home.htm Indiana State University]<br />
<br />
* [http://www-mat.upc.es/grup_de_grafs Research Group on Graph Theory and Combinatorics - UPC, Spain]<br />
<br />
==Newsletters==<br />
<br />
[[Combinatorial_Mathematics_Society_of_Australasia_Newsletters|Newsletters of the Combinatorial Mathematics Society of Australasia]]<br />
<br />
==Meetings, seminars and talks==<br />
<br />
Links to upcoming talks which may be of interest to researchers in the problem areas covered by Combinatorics Wiki.<br />
<br />
[[Mirka_Miller%27s_Combinatorics_Webinar_Series|Mirka Miller's Combinatorics Webinar Series]]<br />
<br />
==Employment Opportunities== <br />
<br />
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.<br />
<br />
<br />
==Combinatorics Wiki rules==<br />
<br />
Please read our rules of [[Rules and Regulations|use of Combinatorics Wiki]]. <br />
<br />
<br />
==Note for potential new contributors and moderators==<br />
<br />
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).</div>
Guillermo