Largest Known (Degree, Diameter)-Graphs

Diameter 5

Last modification: October 14, 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.
semidirect product Follows the notation of the original contributions from Dinneen and Hafner (arXiv preprint) Networks, 24 (1994) 359-367: Zm ⋉a Zn, of order m·n. Group operation defined as [x,y] × [u,v] = [(x+u) mod m, (y·au+v) mod n]].
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

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

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).
--------------
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)∪E(Cox). If v∈V(Cox), u∈V(Cox), then they are not connected in H. If v∈V(Cox), u∈E(Cox), then they are connected in H iff v∈u in G. If v∈E(Cox), u∈E(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
--------------
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
--------------
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).
--------------
This SageMath script, computes several of its properties. This is the online version. .
Rithwik_650

Degree= 5, Diameter = 5; Order=650; Moore bound=1706;

Rithwik Gupta (rithwikca2020@gmail.com ); September 8, 2026). Graph found through AI-assisted exhaustive computational searches (details to come).
This SageMath script computes 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= 648. Cayley graph
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.
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. Coset graph from ? = PSL(2,23) (order 6072) acting on the 1518 cosets of a subgroup of order. 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.
Dhar_2810

Degree= 7, Diameter = 5; Order =2810; Moore bound=10886.

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 7, diameter 5, and 2810 vertices is constructed as a two-copy voltage lift over the semidirect product Z5 ⋉86Z281, of order 1405. The group operation is defined by [x,y] × [u,v] =[(x+u) mod 5, (y·86u+v) mod 281]. The construction uses two copies of the group, with inverse-pair generators [1,6] and [2,220] for the first and second copies, respectively, and five generators [0,242], [2,143], [3,106], [1,115], and [0,251] connecting the two copies. Each connecting generator also defines an inverse edge in the opposite direction. The resulting graph has 2·1405 = 2810 vertices.

Download the raw adjacency list of the graph.
This SageMath script computes 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=2758
Obtained by Michel de Messieres (michel@demessieres.com, October 5, 2026). The search programs were written and run by an AI assistant (Anthropic's Claude) under the author's direction. Lift of a voltage graph with two vertices (dipole) over the semidirect product of cyclic groups Z7 (⋊191) Z197. Each vertex type has six neighbours of its own type and 1 of the other type. The automorphism group is the lifted group, with two orbits, so the graph is not vertex-transitive. They were found by a computer search over such groups of order just above half the former entry , optimising the voltages by simulated annealing until every pair of vertices was within distance 4.
The graph and a detailed write-up are also on Zenodo https://doi.org/10.5281/zenodo.23167440
Download the edge list of the graph.
This SageMath script computes several properties of the graphs. This is the online version .

Former result, order=2756
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
--------------
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_5334

Degree= 8, Diameter = 5; Order =5334; Moore bound=22409

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 the semidirect product Z21 (⋊25) Z254 with generators [2,60 ]<>[19,6 ]:[3,122 ]<>[18,160 ]:[6,81 ]<>[15,241 ]:[8,169 ]<>[13,59 ]. .
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.
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= 5253
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
Bhakar_9020

Degree= 9, Diameter = 5; Order =9020; Moore bound=42130.

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.

This Cayley graph is constructed over the semidirect product Z20 ⋉46 Z451, of order 9020. 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 20, (y·46u+v) mod 451]. The graph has four inverse-pair generators (19,402), (14,161), (13,302), and (11,412), corresponding to the integers 8971, 6475, 6165, and 5373, respectively, together with the involution (10,0), corresponding to the integer 4510.

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=8856
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.
This Cayley graph is constructed over the semidirect product Z24 ⋉14 Z369, with a group operation defined as in the original contributions from M.J. Dinneen and P. Hafner (see top of this page). Generators: [3,312 ]<>[21,186 ]:[18,322 ]<>[6,218 ]:[1,107 ]<>[23,335 ]:[18,174 ]<>[6,33 ]:[12,324 ]<>[12,324 ]. Distance distrib. (1,9,72,576,3637,4561), Avg.dist. 4.430717
Download the local edge list of the graph.
This SageMath script computes several properties of the graph. This is the online version .

Former result, order=8802
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 Z52 (⋊2) Z159, quotient B(1,4), voltages: (14,41), (47,112), (37,82), (10,113), (26,147)"
A. Rodri­guez de los Santos (2013.). Búsquedas masivas de grafos de gran orden con grado y diámetro acotados. Tesis de maestría. Universidad de la República (Uruguay). Facultad de Ingeniería. (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
--------------
This SageMath script, computes several of their properties. This is the online version. .
Rajiv_33397

Degree= 10, Diameter = 5; Order = 33397; Moore bound= 73811.

Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 23, 2026) with the use of AI agentic coding systems. Obtained from the split Cayley hexagon H(9). H(9) has no polarity, but it has dualities of order four. The quotient of its incidence graph (132,860 vertices) by such a duality has 33,397 vertices, maximum degree 10 and diameter 5. These dualities generate conjugate subgroups, so the quotient is unique up to isomorphism. Adding edges makes it 10-regular without changing the diameter.
In this link https://doi.org/10.5281/zenodo.22907863 you can download the edge list, a paper and information detailing the construction of the graph.
Download the local edge list of the graph.

Former result, order= 14047.
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
Dhar_37628

Degree= 11, Diameter =5; Order = 37628; Moore bound = 122221.

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.
Partial fold of the split Cayley hexagon H(9), on the construction and duality data of R. Rajiv: a closed split set, 29 copies of H(3) and 122 pencil vertices give a graph on 37622 vertices, to which 6 further vertices are added, each joined to vertices that still had a free degree slot (50 new edges). No vertex or edge of the 37622-vertex graph is removed.
Download the raw adjacency list of the graph.

Former result, order= 36563.
Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 30, 2026) with the use of AI agentic coding systems. From a partial fold of the split Cayley hexagon H(9). .
In this link https://doi.org/10.5281/zenodo.22907863 you can download the edge list, a paper and information detailing the construction of the graph.
Download the local edge list of the graph.

Former result, order= 35203.
Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 23, 2026) with the use of AI agentic coding systems. Obtained from the (10,5)=33397 graph by adding 1806 twin vertices (new vertices joined to exactly the neighbours of an existing vertex to give the graph of degree 11. .
In this link https://doi.org/10.5281/zenodo.22907863 you can download the edge list, a paper and information detailing the construction of the graph.
Download the local edge list of the graph.

Former result, order= 20952.
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
Dhar_42342

Degree= 12, Diameter =5; Order = 42342; Moore bound = 193260.

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.
Partial fold of the split Cayley hexagon H(9), on the construction and duality data of R. Rajiv: a closed split set with copies of H(3), pencil vertices, and a further 165 vertices each joined to at most 12 vertices that still had a free degree slot, chosen so that the balls of radius 4 around their neighbours cover the graph.
Download the raw adjacency list of the graph.

Former result, order= 41699.
Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 30, 2026) with the use of AI agentic coding systems. From a partial fold of the split Cayley hexagon H(9). .
In this link https://doi.org/10.5281/zenodo.22907863 you can download the edge list, a paper and information detailing the construction of the graph.
Download the local edge list of the graph.

Former result, Order =38167
Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 23, 2026) with the use of AI agentic coding systems. Obtained from the (10,5)=33397 graph by adding 4770 t twin vertices (new vertices joined to exactly the neighbours of an existing vertex to give the graph of degree 12.
In this link https://doi.org/10.5281/zenodo.22907863 you can download the edge list, a paper and information detailing the construction of the graph.
Download the local edge list of the graph.

Former result, order= 34992.
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
Dhar_51417

Degree= 13, Diameter =5; Order = 51417; Moore bound = 294073.

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.
Partial fold of the split Cayley hexagon H(9), on R. Rajiv's (13,5) split set: the colours and the copies of H(3) are chosen jointly, the pencil vertices carry extra anchors with reserved slots, and a packing of twin vertices fills the slots that remain.
Download the raw adjacency list of the graph.

Former result, order= 50750.
Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, September 30, 2026) with the use of AI agentic coding systems. From a partial fold of the split Cayley hexagon H(9). .
In this link https://doi.org/10.5281/zenodo.22907863 you can download the edge list, a paper and information detailing the construction of the graph.
Download the local edge list of the graph.

Former result, Order = 44232
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 graphh generators [1,992 ]<>[23,906 ]:[9,1133 ]<>[15,1166 ]:[10,820 ]<>[14,1259 ]:[12,57 ]<>[12,57 ]:[14,1259 ]<>[10,820 ]:[15,1166 ]<>[9,1133 ]:[23,906 ]<>[1,992 ] .
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= 43890.
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
SamSar_94186

Degree= 14, Diameter =5; Order = 94186; 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 October 22, 2026 (bargainlandz@gmail.com )
Graph obtained by modifying and augmenting Mizuno?s graph (see below).
Download the local edge list .

Former result, order = 88452
Construction discovered by Ryosuke Mizuno (mizuno@aisystemresearch.com, September 25, 2026) through interaction with ChatGPT via its standard web interface. A graph on C × F53, where C is a set of 364 states with four labeled involutions, some of which fix states. This generalizes the controller used in [arXiv:2606.15860] to allow transitions from a state to itself and adds translation edges within each fiber. After removing loops and repeated edges, the graph is simple, has maximum degree 14 and diameter exactly 5; 756 vertices have degree 13 and 87,696 have degree 14.
More details in Mizuno's "degree16-diameter3" github folder
Download the local edge list of the graph.

Former result, order = 61307
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
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 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
Download the implicit adjacency list .

Former result, order =60680
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 Z40 (⋊142) Z1517 with generators [5,174 ]<>[35,399 ]: [5,450 ]<>[35,38 ]: [4,1237 ]<>[36,317 ]: [32,1451 ]<>[8,843 ]: [31,779 ]<>[9,123 ]:[6,100 ]<>[34,1358 ]: [15,861 ]<>[25,615 ].
Ord.: 60680 / Size: 424760 / Diam.: 5 / Avg.dist: 4.51870 / 14-reg.? True / Girth: / dist. distrib. [[1, 14, 182, 2351, 23896, 34236] / Cayley ? True -
Download the implicit adjacency list .

Former result, order =60390
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 16, 2026 (bargainlandz@gmail.com )
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
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_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
SamSar_100132

Degree= 15, Diameter = 5; Order =100132; Moore bound=620565

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 October 22, 2026 (bargainlandz@gmail.com )
Graph obtained by modifying, augmenting and extending Mizuno?s graph (see below).
Download the local edge list .

Former result, order = 100000
Construction discovered by Ryosuke Mizuno (mizuno@aisystemresearch.com, September 25, 2026) through interaction with ChatGPT via its standard web interface. A simple 15-regular graph on V(C) × F55, where C is a 32-vertex simple 3-regular controller. This generalizes the route-chart lift construction of [arXiv:2606.15860] to allow a self-inverse label. Nonsingular controllability matrices and pencils of affine hyperplanes establish diameter at most 5 for all 1,024 ordered pairs of controller vertices; the graph has diameter exactly 5.
More details in Mizuno's "degree16-diameter3" github folder
Download the local edge list of the graph.

Former result, order = 82584
< 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]
Download the raw adjacency list .

Former result, order = 80704
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=80190
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 19, 2026 (bargainlandz@gmail.com )
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
Download the implicit adjacency list .

Former result, Order = 79152
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_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;

Construction discovered through interaction with ChatGPT via its standard web interface.
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
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, (⊗ 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) 325–334. link to the paper