Largest Known (Degree, Diameter)-Graphs

Diameter 8

Last modification: September 1, 2026.
https://web.mat.upc.edu/francesc.comellas/old-files/delta-d/taula_delta_d.html
raw adjacency list format: the first vertex of each row is adjacent to all the other vertices in that row.
implicit adjacency list format: each row corresponds to a vertex (row 1, vertex 0; row 2, vertex 1; and so on) and contains all vertices adjacent to it.
adjacency list NX NetworkX format. NetworkX format.
Chen_360
Degree = 3, Diameter = 8; Order =360; Moore bound=766;
Graph found by Jianxiang Chen (2018-10-16)
The graph is derived from the symmetric graph on 144 vertices with diameter 7 and girth 8 by a complete pairing of its edges that has a large symmetric group. Let G be the symmetric graph and ~ the pairing relation on its edges. The graph is constructed as follows: The vertex set of the new graph H is V(G)∪E(G). If v∈V(G), u∈V(G), then they are not connected in H. If v∈V(G), u∈E(G), then they are connected in H iff v∈u in G. If v∈E(G), u∈E(G), then they are connected in H iff v~u by the pairing relation. The graph H is not a Cayley graph. It has 3 vertex orbits.
Download the raw adjacency list of the graph.
Download the adjacency list (NetworkX) of the graph.
Download a SageMath program to generate the graph in sparse6 format. This SGM program online .
Download a Python program to check the graph. Can be easily adapted to check graphs in all other formats on this website. This Python program online
Download a SageMath script to check several properties of the graph. This is an online version.
Loz_3243
Degree= 4, Diameter = 8; Order =3 243; Moore bound=13121.
Communicated by Eyal Loz, Math Dep., Auckland Univ., New Zealand (July 2006)
Download the raw adjacency list of the graph.
Download a SageMath script to check several properties of the graph. This is an online version.
Link to Eyal Loz's data. (eloz002 @ math .auckland. ac. nz ) Communicated July 2006.
Dahr_17602
Degree= 5, Diameter = 8; Order =17602; Moore bound=109226
Dharunish Yugeswardeenoo (dharyugi@gmail.com; August 17, 2026). Results obtained by using a hybrid discovery framework in development: LLM-guided search over classical algebraic constructions, combined with local search and exact completion. The specific LLMs used were OpenAI's Sol 5.6 and Anthropic's Claude Fable 5 Max. Voltage lift of a 3-vertex base multigraph over the semidirect group Z_677 (x362) Z_13. Order: 17602 / Size: 44005 / 5-reg.? True / Girth: 9 / Diam.: 8 / Avg.dist: 6.70661 / Aut.group.ord.: 8801 / Cayley ? False --- vtx.trans. ? False -- edge.trans. ? False / Automorphism group structure / C677 x| C13
This SageMath online script computes several properties of the graph including symmetry group sizes (run it locally).
Download the raw adjacency list .

Former result, order =17030
Communicated by Eyal Loz, Math Dep., Auckland Univ., New Zealand (July 2006)
Download the raw adjacency list of the graph.
Link to Eyal Loz's data. (eloz002 @ math .auckland. ac. nz ) Communicated July 2006.
Rajiv_77343
Degree= 6, Diameter =8; Order = 77343; Moore bound = 585936.
Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, August 24, 2026) with the use of AI agentic coding systems.
Cayley graph of the semidirect group Z21 (⋊1011) Z3683, with generators [16,2082 ]<>[5,294 ]:[14,2934 ]<>[7,3562 ]:[4,2556 ]<>[17,662 ]. Avg.dist: 6.8410586 . Dist. distrib. :[ 1, 6, 30, 150, 750, 3702, 16540, 41470, 14694 ] .
In this link you can download the adjacency list, a verifier (standard-library Python), and a paper describing the construction of the graph.

Former result, order=77257.
Dharunish Yugeswardeenoo (dharyugi@gmail.com; August 17, 2026). Results obtained by using a hybrid discovery framework in development: LLM-guided search over classical algebraic constructions, combined with local search and exact completion. The specific LLMs used were OpenAI's Sol 5.6 and Anthropic's Claude Fable 5 Max. Voltage lift of a 1-vertex base multigraph over the semidirect group Z3359 (⋊3051) Z23.
Download the raw adjacency list .

Former result, order =76891
Cayley graph. Found as a semidirect product, 76891 nodes and 230673 edges (F.Comellas 2024):
** Z_17 x(891) Z_4523 generators [6,1326]<>[11,4282]:[4,1336]<>[13,119]:[14,1686]<>[3,3044].
Degree Distribution: [0, 0, 0, 0, 0, 0, 76891] Avg. dist.: 6.843725 (1,6,30,150,750,3710,16424,40956,14864) transm.: 526214 adjlist NX
** Z_17 x(1015) Z_4523 generators [16,343]<>[1,126]:[10,615]<>[7,4094]:[12,3868]<>[5,3995].
Degree Distribution: [0, 0, 0, 0, 0, 0, 76891] Avg. dist.: 6.843725 (1,6,30,150,750,3710,16424,40956,14864) transm.: 526214 adjlist NX
** Z_17 x(4359) Z_4523 generators [14,3786]<>[3,4338]:[12,587]<>[5,1066]:[13,572]<>[4,3253].
Degree Distribution: [0, 0, 0, 0, 0, 0, 76891] Avg. dist.: 6.838314 (1,6,30,150,750,3720,16408,41374,14452) transm.: 525798 adjlist NX
Graphs 1 and 2 are isomorphic and have a higher transmission (and thus average distance) than graph 3.
Former result, order = 76461
Communicated by Eyal Loz, Math Dep., Auckland Univ., New Zealand (July 2006)
Voltage graph Z_33 x(79) Z_2317, B(0,3), voltages [(40,5722)(12,5970)(44,3528)]
Z_m x(a) Z_n represents a semidirect product of cyclic groups [x,y][u,v]= [x + u mod m, y*a^u + v mod n].
::: Link to Eyal Loz's original data.
E. Loz, J. Širáň. New record graphs in the degree-diameter problem. Australas. J. Combin. 41 (2008), 63–80.
Degree= 6, Diameter = 8; Order =74 256; Moore bound=585937.
Rajiv_254268
Degree= 7, Diameter =8; Order =254268; Moore bound = .
Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, August 24, 2026) with the use of AI agentic coding systems.
Cayley graph of the semidirect group Z84 (⋊341) Z3027, with generators [1,1326 ]<>[83,1878 ]:[39,1977 ]<>[45,2373 ]:[45,2373 ]<>[39,1977 ]:[83,1878 ]<>[1,1326 ]. Avg.dist: 6.933853. Dist. distrib.: [ 1, 7, 42, 252, 1500, 8841, 47085, 142825, 53715 ] .
In this link you can download the adjacency list, a verifier (standard-library Python), and a paper describing the construction of the graph.
Rajiv_735105
Degree= 8, Diameter =8; Order = 735105; Moore bound = 7686400 .
Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, August 24, 2026) with the use of AI agentic coding systems.
Cayley graph of the semidirect group Z35 (⋊268) Z21003, with generators [1,19391 ]<>[34,7216 ]:[11,8628 ]<>[24,3102 ]:[20,696 ]<>[15,17478 ]:[28,18332 ]<>[7,16117 ]. Avg.dist: 7.00499928 . Dist. distrib. :[ 1, 8, 56, 392, 2726, 18772, 118058, 425734, 169358] .
In this link you can download the adjacency list, a verifier (standard-library Python), and a paper describing the construction of the graph.
Rod_1697688
Degree= 9, Diameter = 8; Order =1 697 688 ; Moore bound=2 929 686
Communicated by A. Rodr­íguez de los Santos, November 16, 2012.
Voltage graph Z_72 x(1413) Z_23579, B(1,4), voltages [(8,5958)|(27,6086)|(37,22093)|(33,22621)|(36,2717], avg. dist.: 7.002767,(1,9,72,576,4608,36503,263761,1034151,358007), transm.: 11888507
Z_m x(a) Z_n represents a semidirect product of cyclic groups [x,y][u,v]= [x + u mod m, y*a^u + v mod n].
A. Rodrí­guez de los Santos, B&iucute;squedas masivas de grafos de gran orden con grado y diámetro acotados. Tesis de maestrí­a (2013). Universidad de la República (Uruguay). Facultad de Ingeniera.
Download the implicit adjacency list (zipped) of the graph (53.3 MB).
This C program generates the graph as the semidirect product Z_72 x(1413) Z_23579 (no voltages) and checks its diameter and vertex degree.
Results for diameter 8 and degrees 10,11,13, 15 and 16 obtained by Eyal Loz and Jozef Širáň. New record graphs in the degree-diameter problem. Australas. J. Combin. 41 (2008), 63--80. Results for diameter 8 and degrees 12 and 14 found by Dharunish Yugeswardeenoo (dharyugi@gmail.com; August 2026),