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

Largest Known (Degree, Diameter)-Graphs

Diameter 5

Last modification: August 27, 2026.

https://web.mat.upc.edu/francesc.comellas/old-files/delta-d/desc_g/desc_g5.html

row 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.

vC, AFY

Degree= 3, Diameter= 5; Order=70; Moore bound=94;

Graph obtained by connecting seven clusters like the one shown in the figure with edges as

\(\large A_{i,j } \leftrightsquigarrow B_{i \pm 2^j, j \pm 1}, \;\; i = 0..6 \;\; and \;\; j = 0..2 \)

[vCo83] C. von Conta. Torus and other networks as communication networks with up to some hundred points. IEEE Trans Comp,c-32 (1983), pp. 657-666.
[AlFiYe86] I. Alegre, M.A. Fiol and , J.L.A Yebra. Some large graphs with given degree and diameter. J. Graph Theory,10 (1986), pp.219-224.

Download the adjacency list of the graph in NetworkX format (vertices notation as in AlFiYe86).

Download the adjacency list of the graph in NetworkX format (vertices labeled 0,2,...69).
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The same graph can be constructed (Francesc Comellas, 2024) following the procedure from Jianxiang Cheng for the (3,8)=360 graph:

The graph is derived, in this case, from the Coxeter symmetric graph on 28 vertices (diameter 4, girth 7) by a complete pairing of its edges.

Let Cox be the Coxeter graph and ~ the pairing relation on its edges (0,1)->(19,20).

The graph is constructed as follows: The vertex set of the new graph H is V(Cox)âˆªE(Cox). If vâˆˆV(Cox), uâˆˆV(Cox), then they are not connected in H. If vâˆˆV(Cox), uâˆˆE(Cox), then they are connected in H iff vâˆˆu in G. If vâˆˆE(Cox), uâˆˆE(Cox), then they are connected in H iff v~u by the pairing relation.

Download the adjacency list of the graph in NetworkX format (vertices labeled 0,2,...69).

Download a SageMath program to generate the graph in sparse6 format. This SGM program online

Download a SageMath program to generate multiple copies of 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

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The same graph can also be constructed (Francesc Comellas, 2026) with a similar procedure from the Bosák graph (28 vertices, diameter 4, girth 3, but not vertex-symmetric nor edge-symettric).

Download a SageMath program to generate multiple copies of the graph in sparse6 format. This SGM program online
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This SageMath script computes several properties of the graph including symmetry group sizes and the number of k-cycles (k=3..10). This is the online version . More information at The House of Graphs.

H′₃

Degree= 4, Diameter = 5; Order=364; Moore bound=485;

Quotient by a polarity of the incidence graph of the regular bipartite generalized hexagon H₃′.

Obtained by C. Delorme (1981) - see pag.11 of J-C. Bermond, B. Bollobas. The Diameter of Graphs: a survey. Congr. Numer. 32 (1981) pp. 3-27, Boca Raton, US.

C. Delorme, G. Farhi. Large graphs with given degree and diameter. IEEE Trans, Comput. c-33 (1984), pp. 857-860.

C. Delorme. Grands graphes de degré et diamètre donnés. European J. Combin. 6 (1985), pp. 291-302.

Download the adjacency list of the graph in NetworkX format (thanks to Vlad Pelekhaty).

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

Conder_648

Degree= 5, Diameter = 5; Order=648; Moore bound=1706;

Communicated by Marston Conder ( m.conder@auckland.ac.nz ) on January 18, 2026 (paper in preparation).

Download a SageMath script to generate this graph. This is the online version.

Download the raw adjacency list of the graph.

Download the adjacency list (NetworkX) of the graph.

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This SageMath script computes several properties of the graph including symmetry group sizes and the number of k-cycles (k=3..). This is the online version .

Rajiv_1518

Degree= 6, Diameter = 5; Order =1518; Moore bound=4687

Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, August 24, 2026) with the use of AI agentic coding systems.

Graph with 1518 nodes and 4554 edges. Degree Distribution: [0, 0, 0, 0, 0, 0, 1518]. Avg. dist.: 4.327620. Aut.group.ord.: 6072. Automorphism group structure: PSL(2,23) | center order: 1.

In this link you can download the adjacency list, a verifier (standard-library Python), and a paper describing the construction of the graph.

This SageMath script gives several properties of the graphs including symmetry group sizes and the number of k-cycles (k=3...). This is the online version .

Former result, order= 1404

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

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

This SageMath script computes several properties of the graph including symmetry group sizes and the number of k-cycles (k=3..). This is the online version .

Former result, order= 1395

Obtained as a Cayley graph for semidirect product Z₄₅ (⋊₉) Z₃₁, generators 40,22 5,8 26,20 19,3 6,20 39,13

M. Sampels (Sampels@Informatik.Uni-Oldenburg.de). Large networks with small diameter.

DH_2756

Degree= 7, Diameter = 5; Order =2756; Moore bound=10886.

Obtained as a Cayley graph for semidirect product of Zm with Zn

Z₅₂ (⋊₂) Z₅₃ with generators [25,45]<>[27,37] (order 52); [30,23]<>[22,18] (order 26); [40,39]<>[12,51] (order 13); [26,0] (order 2) . Avg.dist.: 4.332123 transm.: 11935 adjlist NX

Dinneen,M.J. & Hafner,P.; New results for the degree/diameter problem. Networks, 24 (1994) 359-367.

Other instances (F.Comellas 2024):

Z₅₂ (⋊₈) Z₅₃ with generators [25,45]<>[27,37];[30,23]<>[22,18]; [40,39]<>[12,51];[26,0]. Avg. dist. 4.332123 transm.: 11935 adjlist NX

Z₅₂ (⋊₈) Z₅₃ with generators [47,30]<>[5,4]; [42,51]<>[10,21]; [7,38]<>[45,50]; [26,8]. Avg. dist. 4.332123 transm.: 11935 adjlist NX

Z₅₂ (⋊₁₂) Z₅₃ with generators [36,26]<>[16,50]; [38,32]<>[14,19]; [45,22]<>[7,17]; [26,22]. Avg. dist. 4.341198 transm.: 11960 adjlist NX

Z₅₂ (⋊₁₈) Z₅₃with generators [23,8]<>[29,16]; [38,41]<>[19,39]; [16,39]<>[36,51]; [26,13]. Avg. dist.4.332123 transm.: 11935 adjlist NX

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This SageMath script computes several properties of the graph including symmetry group sizes and the number of k-cycles (k=3..). This is the online version .

Dhar_5253

Degree= 8, Diameter = 5; Order =5253; Moore bound=22409 < DT>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. Cayley graph of the semidirect group Z₅₁ (⋊₂) Z₁₀₃. Generators: [12,90 ]<>[39,81 ]:[42,70 ]<>[9,4 ]:[28,31 ]<>[23,33 ]:[49,2 ]<>[2,95 ]. avg. dist.: 4.3982486. dist. distrib [1, 8, 56, 386, 2184, 2618]

Download the raw adjacency list .

This SageMath script, computes several of properties of the graph raphs. .

Former result, order= 5219

Dharunish Yugeswardeenoo (dharyugi@gmail.com; August 15,2026). 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 semidirect group Z_17 (x114) Z_307. Generators: [1,204 ]<>[16,79 ]:[3,195 ]<>[14,94 ]:[5,239 ]<>[12,70 ]:[8,90 ]<>[9,78 ]. avg. dist.: 4.403986. dist. distrib [1, 8, 56, 392, 2126, 2636]

Download the edge list .

This SageMath script, computes several of properties of the graphs. .

Former result, order= 5115

(8,5)=5115. Ten non-isomorphic Cayley graphs. Found as a semidirect product, 5115 vertices, 20460 edges. (F. Comellas 2024).

** Z_15 x(20) Z_341, generators [2,107]<>[13,108]:[12,80]<>[3,57]:[11,236]<>[4,294]:[5,13]<>[10,152]. Avg. dist.:4.393821 (1,8,56,384,2132,2534) transm.: 22470. adjlist NX

** Z_15 x(49) Z_341, generators [1,56]<>[14,145]:[4,220]<>[11,319]:[13,75]<>[2,134]:[8,17]<>[7,312]. Avg. dist.: 4.382088 (1,8,56,392,2176,2482) transmission: 22410. Cycle distribution: (0, 0, 0, 0, 51150, 312015, ...). adjlist NX

** Z_15 x(69) Z_341, generators [7,100]<>[8,116]:[11,31]<>[4,217]:[14,182]<>[1,59]:[12,327]<>[3,59]. Avg. dist.: 4.380915 (1,8,56,390,2186,2474) ) transmission: 22404. adjlist NX

** Z_15 x(71) Z_341, generators [41,113]<>[1,161]:[10,181]<>[5,94]:[11,57]<>[4,224]:[2,176]<>[13,242]. Avg. dist.: 4.383262 (1,8,56,386,2182,2482) transm.: 22416 . adjlist NX

** Z_15 x(113) Z_341, generators [4,218]<>[11,303]:[11,113]<>[4,285]:[8,321]<>[7,81]:[1,166]<>[14,249]. Avg. dist.: 4.380524 (1,8,56,392,2184,2474) transm.: 22402. adjlist NX

** Z_15 x(196) Z_341, generators [7,2]<>[8,313]:[14,38]<>[1,54]:[5,202]<>[10,106]:[7,116]<>[8,81]. Avg. dist.: 4.382088 (1,8,56,392,2176,2482) transm.: 22410. Cycle distribution: (0, 0, 0, 0, 71610, 281325, ..) adjlist NX

** Z_15 x(235) Z_341, generators [2,239]<>[13,335]:[3,191]<>[12,332]:[5,42]<>[10,255]:[3,19]<>[12,158]. Avg. dist.: 4.380524 (1,8,56,388,2192,2470) transmission: 22402. adjlist NX

** Z_15 x(267) Z_341, generators [13,277]<>[2,257]:[1,290]<>[14,50]:[4.21]<>[11,69]:[10,258]<>[5,105]. Avg. dist.: 4.378178 (1,8,56,392,2196,2462) transm.: 22390. adjlist NX

** Z_15 x(267) Z_341, generators [14,162]<>[1,53]:[6,175]<>[9,114]:[8,18]<>[7,91]:[9,340]<>[6,157]. Avg. dist.:4.390301 (1,8,56,388,2142,2520) transm.: 22452. adjlist NX

** Z_15 x(324) Z_341, generators [14,24]<>[1,67]:[8,7]<>[1,177]:[6,118]<>[9,148]:[13,168]<>[2,211]. Avg. dist.: 4.409855 (1,8,56,384,2050,2626) transmission: 22552. adjlist NX

Marosi_8802

Degree= 9, Diameter = 5; Order =8802; Moore bound=42130.

Graph obtained by Mark Marosi (August 17, 2026) Department of Measurement and Information Systems, Budapest University of Technology and Economics (BME), Budapest, Hungary (marosi@mit.bme.hu). Marosi used Claude (Anthropic) for the search. The graph is a Cayley graph for the semidirect product Z₅₄ (⋊₁₄₁) Z₁₆₃ with generators [33,15 ]<>[21,39 ]:[19,130 ]<>[35,76 ]:[14,68 ]<>[40,15 ]:[29,132 ]<>[25,104 ]:[27,28 ]<>[27,28 ]. Ord.: 8802 / Size: 39609 / Diam.: 5 / Avg.dist: 4.43790 / 9-reg.? True / Girth: 6 / dist. distrib. [1, 9, 72, 564, 3567, 4589] / Deg. histogram.: [0, 0, 0, 0, 0, 0, 0, 0, 0, 8802] , /Aut.group.ord.: 8802 / Cayley ? True --- vtx.trans. ? True -- edge.trans. ? False / Automorphism group structure C163 x| C54

This is the online SageMath script which computes some properties

Former result, order 8268

Communicated by Alexis Rodriguez, Fac. Ing., Univ. Republica, Montevideo (June 2012)

"Graph with 8268 vertices, 37206 edges, diameter 5, avg. dist.4.415991 and maximum degree 9. Obtained with the group Z_52 x(2) Z_159, quotient B(1,4), voltages: (14,41), (47,112), (37,82), (10,113), (26,147)"

A. RodrÃguez de los Santos (2013.). BÃºsquedas masivas de grafos de gran orden con grado y diÃ¡metro acotados. Tesis de maestrÃa. Universidad de la RepÃºblica (Uruguay). Facultad de IngenierÃa. (implicit adjacency list from Alexis) --- adjlist NX version.

Found also as a semidirect product. All instances have a lower average distance (F.Comellas 2024):

Z_52 x(41) Z_159 with generators [11,77]<>[41,65]; [29,134]<>[23,125]; [24,103]<>[28,98]; [38,32]<>[14,19]; [26,105] . Avg.dist.: 4.408975 transm.: 36449 adjlist NX

Z_52 x(50) Z_159 with generators [33,82]<>[19,85]; [47,78]<>[5,33]; [6,80]<>[46,157]; [30,113]<>[22,40]; [26,275] . Avg.dist.:4.400992 transm.: 36383 adjlist NX

Z_52 x(71) Z_159 with generators [16,142]<>[36,149]; [21,31]<>[31,1]; [9,90]<>[43,108]; [3,73]<>[49,43]; [26,108] . Avg.dist.: 4.398452 transm.: 36362 adjlist NX

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

Rajiv_14047

Degree= 10, Diameter = 5; Order = 14047; Moore bound= 73811.

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 Z₁₁ (⋊₁₃₅) Z₁₂₇₇, with generators [10,962 ]<>[1,384 ]:[2,27 ]<>[9,1207 ]:[8,1151 ]<>[3,176 ]:[2,390 ]<>[9,124 ]:[7,1138 ]<>[4,489 ]. Graph with 14047 nodes and 70235 edges, Avg.dist: 4.47088. Aut.group.ord.: 14047. Dist. distrib. : [1, 10, 90, 798, 5526, 7622]. .

In this link you can download the adjacency list, a verifier (standard-library Python), and a paper describing the construction of the graph.

This SageMath script gives several properties of the graphs including symmetry group sizes and the number of k-cycles (k=3...). This is the online version .

Former result, order= 13734.

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. Cayley graph of the semidirect group Z₁₈ (⋊₃₈) Z₇₆₃. Generators: (7,642) (11,542) (10,476) (8,630) (16,253) (2,145) (14,699) (4,4) (15,632) (3,9).

Download the raw adjacency list .

Former result, order=13203.

(10,5)=13203. Five non-isomorphic Cayley graphs. Found as a semidirect product, 13203 vertices, 66015 edges (F. Comellas 2024):

** Z_81 x(22) Z_163, generators [49,70]<>[32,131]:[64,134]<>[17,63]:[78,95]<>[3,18]:[45,156]<>[36,80]:[14,90]<>[67.63]. Avg. dist. 4.426299 (1,10,90,804,5656,6642) transm.: 58274 adjlist NX

** Z_81 x(24) Z_163, generators [14,82]<>[67,161]:[73,41]<>[8,3]:[59,115]<>[22,19]:[5,110]<>[76,69]:[69,66]<>[12,160]. Avg. dist. 4.433419 (1,10,90,802,5566,6734) transm.: 58530 adjlist NX

** Z_81 x(34) Z_163, generators [69,53]<>[12,138]:[22,108]<>[59,161]:[18,157]<>[63,25]:[31,0]<>[50,0]:[66,44]<>[15,47]. Avg. dist. 4.426905 (1,10,90,808,5640,6654) transm.: 58444 adjlist NX

** Z_81 x(36) Z_163, generators [73,95]<>[8,124]:[71,122]<>[10,121]:[47,140]<>[34,27]:[57,162]<>[24,133]:[7,114]<>[74,84]. Avg. dist. avg. dist.: 4.436146 (1,10,90,810,5614,6778) transm.: 58566 adjlist NX

** Z_81 x(51) Z_163, generators [67,56]<>[14,3]:[43,106]<>[38,14]:[16,139]<>[65,64]:[72,134]<>[9,124]:[30,117]<>[51,119]. Avg. dist. avg. dist.:4.432056 (1,10,90,796,5596,6710) transm.: 58512 adjlist NX

Marosi_20952

Degree= 11, Diameter =5; Order = 20952; Moore bound = 122221.

Graph obtained by Mark Marosi (August 17, 2026) Department of Measurement and Information Systems, Budapest University of Technology and Economics (BME), Budapest, Hungary (marosi@mit.bme.hu). Marosi used Claude (Anthropic) for the search. The graph is a Cayley graph for the semidirect product Z₂₄ (⋊₇₈₅) Z₈₇₃ with generators [20,465 ]<>[4,561 ]:[19,708 ]<>[5,114 ]:[13,553 ]<>[11,277 ]:[18,763 ]<>[6,587 ]:[0,42 ]<>[0,831 ]:[12,639 ]<>[12,639 ]]. Ord.: 20952 / Size: 115236 / Diam.: 5 / Avg.dist: 4.48289 / 11-reg.? True / Girth: 6 / dist. distrib. [[1, 11, 110, 1081, 8298, 11451] /Aut.group.ord.: 20952 / Cayley ? True --- vtx.trans. ? True -- edge.trans. ? False / Automorphism group structure C97 ⋊ ((C9 ⋊ C3) ⋊C8) | center order: 1. Download the implicit adjacency list .

This is the online SageMath script which computes some properties (run it locally)

Former result, order 20646

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. Cayley graph of the semidirect group Z_18 (x18) Z_1147. Generators: (11,820) (7,239) (6,990) (12,715) (17,260) (1,382) (12,827) (6,351) (10,493) (8,480) (9,930). Download the raw adjacency list .

Former result, order 19980

Cayley graph. Found as a semidirect product by Niklas Wupperfeld ( niklas_wupperfeld@outlook.de ) in August 6, 2026. Obtained with an 'automated research framework' he built, which used Sol 5.6 from OpenAI as the AI agent. :

** Z_36 x(368) Z_555 with generators [10,160 ]<>[26,260 ]:[13,128 ]<>[23,344 ]:[14,516 ]<>[22,486 ]:[17,543 ]<>[19,24 ]:[30,67 ]<>[6,152 ]:[18,321 ]<>[18,321 ]:]. 4.45463 / Girth: 6 / Aut.group.ord.: 19980 / edge.trans. ? False / distance distrib f [1, 11, 110, 1093, 8336, 10429] adjlist NX

Former result, Order =19620

Cayley graph. Found as a semidirect product, 19620 nodes and 107910 edges (F.Comellas 2024):

** Z_36 x(434) Z_545 generators [22,21 ]<>[14,49 ]:[30,484 ]<>[6,416 ]:[22,513 ]<>[14,107 ]:[33,116 ]<>[3,56 ]:[28,421 ]<>[8,134 ]:[18,285 ]<>[18,285 ]. Avg. dist.: 4.444773 (1,11,110,1088,8343,10067) transm.: 87202 adjlist NX

Mizuno_34992

Degree= 12, Diameter =5; Order = 34992; Moore bound = 193260.

Ord.: 34992 / Size: 209952 / Avg.dist: 4.71111 / Degree histogram: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 34992] / Girth: 4

Aut.group.ord.: 18 / Cayley ? False --- vtx.trans. ? False -- edge.trans. ? False

Construction discovered through interaction with ChatGPT via its standard web interface. Details in: Ryosuke Mizuno, New lower bounds for the degree/diameter problem via interaction with a browser-accessible LLM arXiv:2606.15860 [math.GM]. doi: 10.48550/arXiv.2606.15860v1

Download the edge list of the graph.

Former result, Order = 29621

Cayley graph. Found as a semidirect product, 29621 nodes and 177726 edges (F.Comellas 2024):

Z_19 x(1205) Z_1559, generators [4,358 ] [15,963 ] [12,47 ] [9,233 ] :[14,645 ] [12,1195 ] Avg. dist.: 4.471033 (1,12,132,1452,12320,15704) transm.: 132432. adjlist NX

Rajiv_43890

Degree= 13, Diameter =5; Order = 43890; Moore bound = 294073.

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 Z₃₀ (⋊₆₈) Z₁₄₆₃, with generators [1,27 ]<>[29,1312 ]:[6,425 ]<>[24,240 ]:[8,110 ]<>[22,1089 ]:[15,1026 ]<>[15,1026 ]:[22,1089 ]<>[8,110 ]:[24,240 ]<>[6,425 ]:[29,1312 ]<>[1,27 ]. Graph with 43890 nodes and 285285 edges, Avg.dist: 4.51042 . Aut.group.ord.: 14047. Dist. distrib. :[1, 13, 156, 1843, 17281, 24596] .

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= 42900.

Dharunish Yugeswardeenoo (dharyugi@gmail.com; August 20, 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 semidirect group Z₆₀ (⋊₂) Z₇₁₅ . Generators: [10,512]<>[50,357]:[23,547]<>[37,76]:[59,654]<>[1,122]:[29,615]<>[31,255]:[26,321]<>[34,1]:[46,560]<>[14,555]:[30,198]<>[30,198].

Download the raw adjacency list .

Former result, order = 42680

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. Cayley graph of the semidirect group Z_40 (x50) Z_1067. Generators: (16,833) (24,137) (9,343) (31,1000) (7,792) (33,946) (1,588) (39,287) (13,249) (27,263) (29,998) (11,51) (20,209).

Former result, order =40584

Cayley graph. Found as a semidirect product by Niklas Wupperfeld ( niklas_wupperfeld@outlook.de ) in August 2026. Obtained with an 'automated research framework' he built, which used Sol 5.6 from OpenAI as the AI agent. :

**Z_24 x(1436) Z_1691 generators [1,131 ]<>[23,1287 ]:[2,263 ]<>[22,1382 ]:[5,1386 ]<>[19,961 ]:[13,1509 ]<>[11,1483 ]:[15,1650 ]<>[9,136 ]:[19,827 ]<>[5,1469 ]:[12,171]<>[12,171]. Avg. dist: 4.47865 / Girth: 6 / Aut.group.ord.: 40584 / edge.trans. ? False. distance distrib [1, 13, 156, 1848, 16942, 21624] adjlist NX

Former result, order = 40488

Cayley graph. Found as a semidirect product, 40488 nodes and 263172 edges (F.Comellas 2024):

Z_24 x(362) Z_1687 with generators [1,1454] [5,1427] [2,1659] [15,837] [13,1606] [19,1105] [12,1029]. Avg. dist.:4.479784 (1,13,156,1836,16870,21612) transm.: 181373 adjlist NX

Badaoui_61307

Degree= 14, Diameter =5; Order = 61307; Moore bound = 433174.

Obtained by Sami Hani Badaoui and Sarah Ellen Mary Vaughan ( Sydney, New South Wales, Australia ) through an interactive computer-assisted search conducted with GPT-5.6 Sol from OpenAI via the standard ChatGPT web interface,with final candidates checked by exact Python and C++ breadth-first-search verifiers. Communicated August 22, 2026 (bargainlandz@gmail.com )

The graph is a Cayley graph for the semidirect product Z₁₀₁ (⋊₁₂₂) Z₆₀₇ with generators [60,466 ]<>[41,422 ]:[52,127 ]<>[49,420 ]:[38,603 ]<>[63,174 ]:[21,132 ]<>[80,30 ]:[12,242 ]<>[89,213 ]:[40,318 ]<>[61,423 ]:[65,248 ]<>[36,371 ]; Dist. distribution: [ 1, 14, 182, 2330, 23640, 35140], avg. distance : 4.5284877

Download the implicit adjacency list .

Former result, order = 60705

The graph is a Cayley graph for the semidirect product Z₄₅ (⋊₅₂₂) Z₁₃₄₉ with generators [3,466 ]<>[42,631 ]:[21,182 ]<>[24,183 ]:[11,1249 ]<>[34,856 ]:[7,1282 ]<>[38,335 ]:[19,973 ]<>[26,312 ]:[14,1259 ]<>[31,901 ]:[8,913 ]<>[37,428 ]; Dist. distribution: [ 1, 14, 182, 2346, 23714, 34448], avg. distance : 4.52206572

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Former result, order =60680

Ord.: 60680 / Size: 424760 / Diam.: 5 / Avg.dist: 4.51870 / 14-reg.? True / Girth: / dist. distrib. [[1, 14, 182, 2351, 23896, 34236] / Cayley ? True -

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Former result, order =60390

Z_45 x(423) Z_1342 with generators [3,83 ] [6,1120 ] [11,40 ] [13,1130 ] [19,973 [20,358 ] [6,209 ]

Former result, order =59085

Z_45 x(1004) Z_1313 with generators [11,570 ] [16,182 ] [18,866 ] [24,151 ] [32,781 ] [36,615 ] [41,676 ] Avg.dist: 4.51594 / Girth: 6 / Aut.group.ord.: 59085 / edge.trans. ? False; distance distrib: [1, 14, 182, 2336, 23326, 33226] adjlist NX

Former result, Order = 58095

Cayley graph. Found as a semidirect product, 58095 nodes and 406665 edges (F.Comellas 2024):

Z_45 x(191) Z_1291 with generators [31,28 ]<>[14,108 ]:[32,290 ]<>[13,507 ]:[28,326 ]<>[17,1177 ]:[41,665 ]<>[4,735 ]:[18,278 ]<>[27,397 ]:[24,148 ]<>[21,1269 ]:[36,259 ]<>[9,957 ] Avg. dist.: 4.503047 (1,14,182,2362,23544,31992) transm.: 261600 adjlist NX

Dhar_82584

Degree= 15, Diameter = 5; Order =82584; Moore bound=620565

< DT> Dharunish Yugeswardeenoo (dharyugi@gmail.com; September 4, 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 semidirect group Z₃₆ (⋊₅₇) Z₂₂₉₄. Generators: [5,929 ]<>[31,119 ]:[8,1792 ]<>[28,2076 ]:[19,515 ]<>[17,1219 ]:[22,467 ]<>[14,2151 ]:[7,2210 ]<>[29,690 ]:[16,1994 ]<>[20,990 ]:[13,1984 ]<>[23,1302 ]:[18,2263 ]<>[18,2263 ]. dist. distrib [1, 15, 210, 2916, 31885, 47557]

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Former result, order = 80704

Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, August 24, 2026) with the use of AI agentic coding systems.

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=80190

The graph is a Cayley graph for the semidirect product Z₉₀ (⋊₅₄₈) Z₈₉₁ with generators [29,538 ]<>[61,691 ]:[67,379 ]<>[23,469 ]:[1,846 ]<>[89,522 ]:[25,854 ]<>[65,818 ]:[20,118 ]<>[70,80 ]:[31,500 ]<>[59,140 ]:[4,530 ]<>[86,532 ]:[45,363 ]<>[45,363 ]]; Dist. distribution: [ 1, 15, 210, 2910, 31683, 45371], avg. distance : 4.5236562

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Former result, Order = 79152

** Z_48 x(772) Z_1649 with generators [2,842 ]<>[46,1560 ]:[3,521 ]<>[45,1498 ]:[10,300 ]<>[38,241 ]:[23,875 ]<>[25,396 ]:[28,1107 ]<>[20,791 ]:[31,1066 ]<>[17,1412 ]:[47,314 ]<>[1,1644 ]:[24,0 ]<>[24,0 ]: / Avg.dist: 4.52586 / Girth: 5 / Aut.group.ord.: 79152 / edge.trans. ? False /distance distrib: [1, 15, 210, 2891, 31057, 44978] adjlist NX

Former result, Order = 77520

Cayley graph. Found as a semidirect product, 77520 nodes and 581400 edges (F.Comellas 2024):

Z_48 x(772) Z_1615 generators [3,482]<>[45,406]:[28,1131]<>[20,1239]:[31,682]<>[17,1346]:[47,1424]<>[1,487]:[2,831]<>[46,16]:[10,300]<>[38,785]:[23,1068]<>[25,1339]:[24,0]<>[24,0] . Degree Distribution: [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,77520] Avg. dist.:4.517228 (1,15,210,2877,30980,43437) transm.: 350171 adjlist NX

Mizuno_147456

Degree= 16, Diameter= 5; Order= 147456; Moore bound= 867857;

Ord.: 147456 / Size: 1179648 / Avg.dist: 4.79109 / Degree histogram: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 147456] / Girth: 4

Aut.group.ord.: 36 / Cayley ? False --- vtx.trans. ? False -- edge.trans. ? False

Construction discovered through interaction with ChatGPT via its standard web interface.

Details in: Ryosuke Mizuno, New lower bounds for the degree/diameter problem via interaction with a browser-accessible LLM arXiv:2606.15860 [math.GM]. doi: 10.48550/arXiv.2606.15860v1

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Former result, (⊗ H₃)′ Order = 132496

The component with polarity of the cartesian product of the incidence graph of regular generalized hexagon H_3 by itself

C. Delorme. Large bipartite graphs with given degree and diameter. J. Graph Theory, 9 (1985) 325â€“334. link to the paper