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Table of the largest known graphs of a given diameter and maximal degree

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In graph theory, the degree diameter problem is the problem of finding the largest possible graph for a given maximum degree and diameter. The Moore bound sets limits on this, but for many years mathematicians in the field have been interested in a more precise answer. The table below gives current progress on this problem (excluding the case of degree 2, where the largest graphs are cycles with an odd number of vertices).

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

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Below is the table of the vertex numbers for the best-known graphs (as of July 2022) in the undirected degree diameter problem for graphs of degree at most 3 ≤ d ≤ 16 and diameter 2 ≤ k ≤ 10. Only a few of the graphs in this table (marked in bold) are known to be optimal (that is, largest possible). The remainder are merely the largest so far discovered, and thus finding a larger graph that is closer in order (in terms of the size of the vertex set) to the Moore bound is considered an open problem. Some general constructions are known for values of d and k outside the range shown in the table.

k
d
 2 3 4 5 6 7 8 9 10
3 10[g 1] 20[g 2] 38[g 3] 70[g 4] 132[g 5] 196[g 5] 360[g 6] 600[g 5] 1250[g 7]
4 15[g 2] 41[g 8] 98[g 5] 364[g 9] 740[g 10] 1 320 3 243 7 575 17 703
5 24[g 2] 72[g 5] 212[g 5] 624 2 772[g 11] 5 516 17 030 57 840 187 056
6 32[g 12] 111[g 5] 390 1404 7 917[g 11] 19 383 76 461 331 387[g 13] 1 253 615
7 50[g 14] 168[g 5] 672[g 15] 2 756[g 16] 11 988 52 768 249 660 1 223 050 6 007 230
8 57[g 17] 253[g 18] 1 100 5 060 39 672[g 19] 131 137 734 820 4 243 100 24 897 161
9 74[g 17] 585[g 9] 1 550 8 268[g 13] 75 893[g 11] 279 616 1 697 688[g 13] 12 123 288 65 866 350
10 91[g 17] 650[g 9] 2 286 13 140 134 690[g 19] 583 083 4 293 452 27 997 191 201 038 922
11 104[g 5] 715[g 9] 3 200[g 20] 19 500 156 864[g 20] 1 001 268 7 442 328 72 933 102 600 380 000
12 133[g 21] 786[g 19] 4 680[g 22] 29 470 359 772[g 11] 1 999 500 15 924 326 158 158 875 1 506 252 500
13 162[g 23] 856[g 24] 6 560[g 20] 40 260 531 440[g 20] 3 322 080 29 927 790 249 155 760 3 077 200 700
14 183[g 21] 916[g 19] 8 200[g 20] 57 837 816 294[g 11] 6 200 460[g 25] 55 913 932 600 123 780 7 041 746 081
15 187[g 26] 1 215[g 27] 11 712[g 20] 76 518 1 417 248[g 20] 8 599 986 90 001 236 1 171 998 164 10 012 349 898
16 200[g 28] 1 600[g 27] 14 640[g 20] 132 496[g 27] 1 771 560[g 20] 14 882 658[g 25] 140 559 416 2 025 125 476 12 951 451 931

Entries without a footnote were found by Loz & Širáň (2008). In all other cases, the footnotes in the table indicate the origin of the graph that achieves the given number of vertices:

  1. ^ The Petersen graph.
  2. ^ a b c Optimal graphs proven optimal by Elspas (1964).
  3. ^ Optimal graph found by Doty (1982) and proven optimal by Buset (2000).
  4. ^ Graph found by Alegre, Fiol & Yebra (1986).
  5. ^ a b c d e f g h i Graphs found by Geoffrey Exoo from 1998 through 2010.
  6. ^ Graph found by Jianxiang Cheng in 2018.
  7. ^ Graph found by Conder (2006).
  8. ^ Graph found by Allwright (1992).
  9. ^ a b c d Graphs found by Delorme (1985a).
  10. ^ Graph found by Comellas & Gómez (1994).
  11. ^ a b c d e Graphs found by Pineda-Villavicencio et al. (2006).
  12. ^ Optimal graph found by Wegner (1977) and proven optimal by Molodtsov (2006).
  13. ^ a b c Graphs found by Canale & Rodríguez (2012).
  14. ^ The Hoffman–Singleton graph.
  15. ^ Graph found by Sampels (1997).
  16. ^ Graph found by Dinneen & Hafner (1994).
  17. ^ a b c Graphs found by Storwick (1970).
  18. ^ Graph found by Margarida Mitjana and Francesc Comellas in 1995, and independently by Sampels (1997).
  19. ^ a b c d Graphs found by Gómez (2009).
  20. ^ a b c d e f g h i Graphs found by Gómez & Fiol (1985).
  21. ^ a b Graphs found by Delorme & Farhi (1984).
  22. ^ Graph found by Bermond, Delorme & Farhi (1982).
  23. ^ McKay–Miller–Širáň graphs found by McKay, Miller & Širáň (1998).
  24. ^ Graph found by Vlad Pelakhaty in 2021.
  25. ^ a b Graphs found by Gómez, Fiol & Serra (1993).
  26. ^ Graph found by Eduardo A. Canale in 2012.
  27. ^ a b c Graphs found by Delorme (1985b).
  28. ^ Graph found by Abas (2016).

References

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  • Abas, Marcel (2016), "Cayley graphs of diameter two with order greater than 0.684 of the Moore bound for any degree", European Journal of Combinatorics, 57: 109–120, arXiv:1511.03706, doi:10.1016/j.ejc.2016.04.008
  • Comellas, Francesc; Gómez, José (1994). "New Large Graphs with Given Degree and Diameter". arXiv:math/9411218.
  • Doty, Karl (1982), "Large regular interconnection networks", Proceedings of the 3rd International Conference on Distributed Computing Systems, IEEE Computer Society, pp. 312–317
  • Elspas, Bernard (1964), "Topological constraints on interconnection-limited logic", 1964 Proceedings of the Fifth Annual Symposium on Switching Circuit Theory and Logical Design, pp. 133–137, doi:10.1109/SWCT.1964.27
  • Gómez, José (2009), "Some new large (Δ, 3)-graphs", Networks, 53 (1): 1–5, doi:10.1002/NET.V53:1
  • Gómez, José; Fiol, Miquel (1985), "Dense compound graphs", Ars Combinatoria, 20: 211–237
  • Molodtsov, Sergey (2006), General Theory of Information Transfer and Combinatorics, pp. 853–857, ISBN 978-3-540-46244-6
  • Sampels, Michael (1997), "Large Networks with Small Diameter", Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 1335, Springer, Berlin, Heidelberg, pp. 288–302, doi:10.1007/BFb0024505, ISBN 978-3-540-69643-8
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