Table of vertex-symmetric digraphs

Table of the orders of the largest known vertex-symmetric graphs for the directed degree diameter problem

Below is the table of the best known vertex transitive digraphs (as of October 2008) in 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

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

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.