Largest Known (Degree, Diameter)-Graphs (Description)

Largest Known (Degree, Diameter)-Graphs

Diameter 8

Last modification: October 22, 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.

semidirect product Follows the notation of the original contributions from Dinneen and Hafner (arXiv preprint) Networks, 24 (1994) 359-367: Z_m ⋉_a Z_n, of order m·n. Group operation defined as [x,y] × [u,v] = [(x+u) mod m, (y·a^u+v) mod n]].

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.

Bhakar_18810

Degree= 5, Diameter = 8; Order =18810; Moore bound=109226

Obtained by Nishant Bhakar (nishant.bhakar@gmail.com, Redmond, WA, USA; October 15, 2026), who set the research, selected the topics to pursue, and reviewed the results. Claude Fable assisted with the computational searches, code development, analysis, and summarization of the results.

A graph with maximum degree 5, diameter 8, and 18810 vertices is constructed as a two-copy voltage lift over the semidirect product Z₄₅ ⋉₄Z₂₀₉, of order 9405 (209 = 11·19). The group operation (following the notation of the original contributions to this table -see top of this page-) is defined by[x,y] × [u,v] =[(x+u) mod 45, (y·4^u+v) mod 209]. The construction uses two copies of the group, labelled 0 and 1, with internal inverse-pair generators (28,0) and (25,51), respectively, corresponding to the integers 5852 and 5276. The copies are connected by three generators: (0,0), (44,157), and (24,166), corresponding to the integers 0, 9353, and 5182, respectively. Each connecting generator also defines its inverse edge in the opposite direction. The resulting graph has 2·9405 = 18810 vertices, is 5-regular, and has diameter 8. The construction was obtained through a systematic symmetry-reduced search of two-vertex voltage lifts over odd-order metacyclic groups.

In this link you can download the edge list and more information detailing the construction of the graph.

Download the local edge list of the graph.

This SageMath script computes several properties of the graphs. This is the online version .

Former result, order=17724

Dharunish Yugeswardeenoo (dharyugi@gmail.com; Chalfont, PA, USA; October 7, 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.

A graph with maximum degree 5, diameter 8, and 17724 vertices is constructed as a two-copy voltage lift over the semidirect product Z₂₁ ⋉₃₉₁Z₄₂₂, of order 8862. The group operation is defined by [x,y] × [u,v] =[(x+u) mod 21, (y·391^u+v) mod 422]. The construction uses two copies of the group, with inverse-pair generators [10,209] and [16,158] for the first and second copies, respectively, and three generators [0,45], [15,29], and [1,392] connecting the two copies. Each connecting generator also defines an inverse edge in the opposite direction. The resulting graph has 2·8862 = 17724 vertices.

Download the local edge list of the graph.

Former result, order = 17602

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.

Bhakar_80050

Degree= 6, Diameter =8; Order = 80050; Moore bound = 585937.

This Cayley graph is constructed over the semidirect product Z₂₅ ⋉₁₄₁₉ Z₃₂₀₂.. The group operation (following the notation of the original contributions to this table -see top of this page-) is defined by [x,y] × [u,v] = [(x+u) mod 25, (y·1419^u+v) mod 3202]. The graph has three inverse-pair generators (24,519), (20,163), and (14,2724), corresponding to the integers 77367, 64203, and 47552, respectively. The graph was obtained through an exhaustive symmetry-reduced search of Cayley graphs over this group.

In this link you can download the edge list and more information detailing the construction of the graph.

Download the local edge list of the graph.

Former result, order=78825

Dharunish Yugeswardeenoo (dharyugi@gmail.com; Chalfont, PA, USA; October 14, 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.

A Cayley graph constructed over the semidirect product Z₇₅ ⋉₁₄ Z₁₀₅₁, of order 78825. The graph has three inverse-pair generators [43,592], [74,877], and [69,80], corresponding to the integers 45785, 78651, and 72599. Distance distribution [ 1,6,30,150,750,3698,16410,41938,15842].

Download the local raw adjacency list of the graph.

Former result, order=78561

Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 24, 2026) with the use of AI agentic coding systems. The graph is a Cayley graph for a semidirect product Z₂₁ (⋊₃₁₀) Z₃₇₄₁, S= {4456,15631,23596,56924,65366,78257}

In this link https://doi.org/10.5281/zenodo.22928337 you can download the edge list, a paper and information detailing the construction of the graph.

Download the local implicit adjacency list of the graph.

Former result, order = 77343.

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.

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.

Current entries for diameter 8 and degrees 8, 9, 10 and 14 found by Dharunish Yugeswardeenoo (dharyugi@gmail.com; Chalfont, PA, USA; October 19, 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. The graphs are Cayley graphs of metacyclic groups arising from semidirect products of cyclic groups. Z_m ⋉_a Z_n.

Download this file with a .json file and other details to reconstruct the graphs. Note that, in these files, the AI used the opposite convention for the group operation in the semidirect product compared to the one preferred in this table, which follows the original contributions of M. J. Dinneen and P. Hafner (see the top of this page).

Current entries for diameter 8 and degrees 7 to 16, except 8, 9,10 and 14, obtained by Nishant Bhakar (nishant.bhakar@gmail.com, Redmond, WA, USA; October 15, 2026), who set the research, selected the topics to pursue, and reviewed the results. Claude Fable assisted with the computational searches, code development, analysis, and summarization of the results. All are Cayley graph constructed over semidirect products.

In this link you can download more information detailing the construction of the graphs. Note that, in these files, the AI used the opposite convention for the group operation in the semidirect product compared to the one preferred in this table, which follows the original contributions of M. J. Dinneen and P. Hafner (see the top of this page).

Former results (up to October 14, 2026)

Entries for diameter 8 and degrees 7 to 16 found by Dharunish Yugeswardeenoo (dharyugi@gmail.com; Chalfont, PA, USA; October 6-14, 2026). See above.

Former results

Results for diameter 8 and degree 13 obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 30, 2026). In this link: https://doi.org/10.5281/zenodo.23057003 you can download the details and a paper describing the construction of the graphs.

Results for diameter 8 and degree 15 found by Chenxudong, Hangzhou, China (qq475519905@gmail.com; September 2026)..

Results for diameter 8 and degrees 12 and 14 found by Dharunish Yugeswardeenoo (dharyugi@gmail.com; August 2026).

Result for diameter 8 and degrees 5 to 16 obtained by Eyal Loz and Jozef Širáň. Details at New record graphs in the degree-diameter problem. Australas. J. Combin. 41 (2008), 63--80.