The best known vertex transitive digraphs (as of October 2008) in the directed Degree diameter problem are tabulated below.
Table of the orders of the largest known vertex-symmetric graphs for the directed degree diameter problem
k d | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 6 | 10 | 20 | 27 | 72 | 144 | 171 | 336 | 504 | 737 |
| 3 | 12 | 27 | 60 | 165 | 333 | 1 152 | 2 041 | 5 115 | 11 568 | 41 472 |
| 4 | 20 | 60 | 168 | 465 | 1 378 | 7 200 | 14 400 | 42 309 | 137 370 | 648 000 |
| 5 | 30 | 120 | 360 | 1 152 | 3 775 | 28 800 | 86 400 | 259 200 | 1 010 658 | 5 184 000 |
| 6 | 42 | 210 | 840 | 2 520 | 9 020 | 88 200 | 352 800 | 1 411 200 | 5 184 000 | 27 783 000 |
| 7 | 56 | 336 | 1 680 | 6 720 | 20 160 | 225 792 | 1 128 960 | 5 644 800 | 27 783 000 | 113 799 168 |
| 8 | 72 | 504 | 3 024 | 15 120 | 60 480 | 508 032 | 3 048 192 | 18 289 152 | 113 799 168 | 457 228 800 |
| 9 | 90 | 720 | 5 040 | 30 240 | 151 200 | 1 036 800 | 7 257 600 | 50 803 200 | 384 072 192 | 1 828 915 200 |
| 10 | 110 | 990 | 7 920 | 55 400 | 332 640 | 1 960 200 | 15 681 600 | 125 452 800 | 1 119 744 000 | 6 138 320 000 |
| 11 | 132 | 1 320 | 11 880 | 95 040 | 665 280 | 3 991 680 | 31 152 000 | 282 268 800 | 2 910 897 000 | 18 065 203 200 |
| 12 | 156 | 1 716 | 17 160 | 154 440 | 1 235 520 | 8 648 640 | 58 893 120 | 588 931 200 | 6 899 904 000 | 47 703 427 200 |
| 13 | 182 | 2 184 | 24 024 | 240 240 | 2 162 160 | 17 297 280 | 121 080 960 | 1 154 305 152 | 15 159 089 098 | 115 430 515 200 |
Key to colors
| Color | Details |
|---|---|
| * | Family of digraphs found by W.H.Kautz. More details are available in a paper by the author. |
| * | Family of digraphs found by V.Faber and J.W.Moore. More details are available also by other authors. |
| * | Digraph found by V.Faber and J.W.Moore. The complete set of cayley digraphs in that order was found by Eyal Loz. |
| * | Digraphs found by Francesc Comellas and M. A. Fiol. More details are available in a paper by the authors. |
| * | Cayley digraphs found by Michael J. Dinneen. Details about this graph are available in a paper by the author. |
| * | Cayley digraphs found by Michael J. Dinneen. The complete set of cayley digraphs in that order was found by Eyal Loz. |
| * | Cayley digraphs found by Paul Hafner. Details about this graph are available in a paper by the author. |
| * | Cayley digraph found by Paul Hafner. The complete set of cayley digraphs in that order was found by Eyal Loz. |
| * | Digraphs found by J. Gómez. |
| * | Cayley digraphs found by Eyal Loz. More details are available in a paper by Eyal Loz and Jozef Širáň. |
References
- Kautz, W.H. (1969), "Design of optimal interconnection networks for multiprocessors", Architecture and Design of Digital Computers, Nato Advanced Summer Institute: 249–272
- Faber, V.; Moore, J.W. (1988), "High-degree low-diameter interconnection networks with vertex symmetry:the directed case", Technical Report LA-UR-88-1051, los Alamos National Laboratory
- J. Dinneen, Michael; Hafner, Paul R. (1994), "New Results for the Degree/Diameter Problem", Networks, 24 (7): 359–367, arXiv:math/9504214, doi:10.1002/net.3230240702
- Comellas, F.; Fiol, M.A. (1995), "Vertex-symmetric digraphs with small diameter", Discrete Applied Mathematics, 58 (1): 1–12, doi:10.1016/0166-218X(93)E0145-O
- Miller, Mirka; Širáň, Jozef (2005), "Moore graphs and beyond: A survey of the degree/diameter problem" (PDF), Electronic Journal of Combinatorics, Dynamic, survey D
- Loz, Eyal; Širáň, Jozef (2008), "New record graphs in the degree-diameter problem" (PDF), Australasian Journal of Combinatorics, 41: 63–80
External links
- Vertex-symmetric Digraphs online table.
- The Degree - Diameter Problem on CombinatoricsWiki.org.
- Eyal Loz's Degree-Diameter problem page.
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