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

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

Below is the table of the vertex numbers for the best-known graphs (as of October 2008) 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
2345678910
3 102038701321963606001250
4 1541983647401 3203 2437 57517 703
5 24722126242 7725 51617 03057 840187 056
6 3211139014047 91719 38376 461331 3871 253 615
7 501686722 75611 98852 768249 6601 223 0506 007 230
8 572531 1005 06039 672131 137734 8204 243 10024 897 161
9 745851 5508 26875 893279 6161 697 68812 123 28865 866 350
10 916502 28613 140134 690583 0834 293 45227 997 191201 038 922
11 1047153 20019 500156 8641 001 2687 442 32872 933 102600 380 000
12 1337864 68029 470359 7721 999 50015 924 326158 158 8751 506 252 500
13 1628516 56040 260531 4403 322 08029 927 790249 155 7603 077 200 700
14 1839168 20057 837816 2946 200 46055 913 932600 123 7807 041 746 081
15 1871 21511 71276 5181 417 2488 599 98690 001 2361 171 998 16410 012 349 898
16 2001 60014 640132 4961 771 56014 882 658140 559 4162 025 125 47612 951 451 931

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

ColorDetails
*The Petersen and Hoffman–Singleton graphs.
*Optimal graphs which are not Moore graphs.
*Graph found by James Allwright.
*Graph found by G. Wegner.
*Graphs found by Geoffrey Exoo.
*McKay–Miller–Širáň graphs found by McKay, Miller & Širáň (1998)
*Graphs found by J. Gómez.
*Graph found by Mitjana M. and Francesc Comellas. This graph was also found independently by Michael Sampels.
*Graph found by Fiol, M.A. and Yebra, J.L.A.
*Graph found by Francesc Comellas and J. Gómez.
*Graphs found by G. Pineda-Villavicencio, J. Gómez, Mirka Miller and H. Pérez-Rosés. More details are available in a paper by the authors.
*Graphs found by Eyal Loz. More details are available in a paper by Eyal Loz and Jozef Širáň.
*Graphs found by Michael Sampels.
*Graphs found by Michael J. Dinneen and Paul Hafner. More details are available in a paper by the authors.
*Graph found by Marston Conder.

References
  • Hoffman, Alan J.; Singleton, Robert R. (1960), "Moore graphs with diameter 2 and 3" (PDF), IBM Journal of Research and Development, 5 (4): 497–504, doi:10.1147/rd.45.0497, MR 0140437
  • Pineda-Villavicencio, Guillermo; Gómez, José; Miller, Mirka; Pérez-Rosésd, Hebert (2006), "New Largest Graphs of Diameter 6", Electronic Notes in Discrete Mathematics, 24: 153–160, doi:10.1016/j.endm.2006.06.044
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.