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

Largest Known (Degree, Diameter)-Graphs

Diameter 8

Last modification: October 14, 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.

Dahr_17724

Degree= 5, Diameter = 8; Order =17724; Moore bound=109226

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.

Dhar_78825

Degree= 6, Diameter =8; Order = 78825; Moore bound = 585937.

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.

Dhar_255938

Degree= 7, Diameter =8; Order =255938; Moore bound = 2351462 .

Voltage lift of a base graph with 2 vertices over the metacyclic group Z₇₃ ⋉₁₀₀₆ Z₁₇₅₃ of order 127969: the vertices are the pairs (copy, group element) and (c,x) is adjacent to (c',x# g) for each of the 7 units (loop pairs inside a copy, edges between copies) listed in the .json file.

Download this file with a .json file and other details to reconstruct the graph. 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 result, order = 254268.

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.

Dhar_744942

Degree= 8, Diameter =8; Order = 744942; Moore bound = 7686400 .

Voltage lift of a base graph with 2 vertices over the metacyclic group Z₃₃ ⋉₆₇₇₆ Z₁₁₂₈₇ of order 372471: the vertices are the pairs (copy, group element) and (c,x) is adjacent to (c',xg) for each of the 8 units (loop pairs inside a copy, edges between copies) listed in the .json file.

Former result, order = 744878.

Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 30, 2026) with the use of AI agentic coding systems Anthropic's Claude Fable 5.1 and Opus 5.5.

Cayley graph for the semidirect product Z₂₃ (⋊₂₄₉₁) Z₃₂₃₈₆ S= {69812, 184915, 266362, 294870, 479750, 502200, 605493, 682624}

In this link: https://doi.org/10.5281/zenodo.23057003 you can download the details and a paper describing the construction of the graph.

Former result, order = 735175.

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.

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.

Former result, order = 735105.

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.

Dhar_1859260

Degree= 9, Diameter = 8; Order = 1859260 ; Moore bound=2 929 686

Voltage lift of a base graph with 2 vertices over the metacyclic group Z₆₅ ⋉₂₇₈₇ Z₁₄₃₀₂ of order 929630: the vertices are the pairs (copy, group element) and (c,x) is adjacent to (c',x g) for each of the 9 units (loop pairs inside a copy, edges between copies) listed in in the .json file.

Former result, order = 1857060.

Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 30, 2026) with the use of AI agentic coding systems Anthropic's Claude Fable 5.1 and Opus 5.5.

Cayley graph for the semidirect product Z₉₀ (⋊₂₃₃) Z₂₀₆₃₄ S= {232439, 393476, 675380, 870223, 938353, 1008533, 1211224, 1472366, 1633979}.

In this link: https://doi.org/10.5281/zenodo.23057003 you can download the details and a paper describing the construction of the graph.

Former result, order = 1698630.

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.

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.

Former result, order = 1697688.

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, Búsquedas 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.

Dhar_4387768

Degree= 10, Diameter = 8; Order =4387768; Moore bound=538084007.

Dharunish Yugeswardeenoo (dharyugi@gmail.com; Chalfont, PA, USA; October 10, 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.

Cayley graph of the metacyclic group Z₈₈ ⋉₈₂₅ Z₄₉₈₆₁. Generators can be found in the corresponding .json file.

Former result, order = 4387740.

Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 30, 2026) with the use of AI agentic coding systems Anthropic's Claude Fable 5.1 and Opus 5.5.

Cayley graph for the semidirect product Z₈₄ (⋊₃₇) Z₅₂₂₃₅ S= {620896, 860419, 1226861, 1565380, 1986157, 2428307, 2888370, 3188413, 3601696, 3858743}

In this link: https://doi.org/10.5281/zenodo.23057003 you can download the details and a paper describing the construction of the graph.

Former result, order = 4293452.

Communicated by Eyal Loz, Math Dep., Auckland Univ., New Zealand (July 2006)

Cayley graph of the semidirect group Z₆₈ (⋊₁₃₇₈) Z₆₃₁₃₉, with generators 40,7668 ]<>[28,61465 ]:[5,16311 ]<>[63,51074 ]:[20,15555 ]<>[48,29103 ]:[52,7399 ]<>[16,23885 ]:[21,22796 ]<>[47,26610 ]: Average distance : 7.08645, Distance distribution: [ 1 , 10, 90, 810, 7290, 65148, 543291, 2606352, 1070460]

Link to Eyal Loz's data. (eloz002 @ math .auckland. ac. nz )

E. Loz, J. Širáň. New record graphs in the degree-diameter problem. Australas. J. Combin. 41 (2008), 63–80.

Dhar_9261072

Degree= 11, Diameter = 8; Order =9261072 ; Moore bound=1222222216 .

Cayley graph of the metacyclic group Z₄₈ ⋉₇₉₆ Z₁₉₂₉₃₉. Generators can be found in the corresponding .json file.

Former result, order = 9260784.

Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 30, 2026) with the use of AI agentic coding systems Anthropic's Claude Fable 5.1 and Opus 5.5. Cayley graph for the semidirect product Z₄₈ (⋊₄₈₈) Z₁₉₂₉₃₃ S= {770788, 1672954, 1984422, 2160410, 2925207, 4707729, 6502959, 7212950, 7411398, 7764836, 8835553}

In this link: https://doi.org/10.5281/zenodo.23057003 you can download the details and a paper describing the construction of the graph.

Former result, order = 8225902.

Chenxudong (Hangzhou, China; qq475519905@gmail.com; September 28, 2026).

Cayley graph of the semidirect group Z₄₇₈ (⋊₈₁₀₃) Z₁₇₂₀₉, with generators [6,13246 ]<>[472,15454 ]:[165,1539 ]<>[313,3877 ]:[273,4559 ]<>[205,11762 ]:[66,3589 ]<>[412,3261 ]:[413,9502 ]<>[65,1117 ]:[239,12035 ]<>[239,12035 ] Average distance : 7.08902671, Distance distribution: [ 1 , 11, 110, 1100, 10964, 108138, 994955, 5129152, 1981471]

Former result, order = 7442328

Link to Eyal Loz's data. (eloz002 @ math .auckland. ac. nz )

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 12 to 16 found by (dharyugi@gmail.com; Chalfont, PA, USA; October 6-14, 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. All graphs are voltage lifts of a base graph with 2 vertices over a metacyclic group 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).

Former results (up to October 14, 2026)

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 degree 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.