Largest Known (Degree, Diameter)-Graphs
Diameter 7
Last modification: August 31, 2026.
http://www-mat.upc.es/grup_de_grafs/desc_g7.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.
Exoo_196
Degree = 3, Diameter = 7; Order =196; Moore bound=382;
Download the implicit 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
Loz_1320
Degree= 4, Diameter = 7; Order =1320; Moore bound=4373.
Communicated by Eyal Loz, Math Dep., Auckland Univ., New Zealand (July 2006)
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
Link to Eyal Loz's data. (eloz002 @ math .auckland. ac. nz )
Loz_5516
Degree= 5, Diameter = 7; Order =5 516; Moore bound=27 306.
Communicated by Eyal Loz, Math Dep., Auckland Univ., New Zealand (July 2006)
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
Link to Eyal Loz's data. (eloz002 @ math .auckland. ac. nz ) .
Com_20319
Degree= 6, Diameter = 7; Order =20319; Moore bound= 117186
Cayley graph. Found as a semidirect product, 20319 nodes and 60957 edges (F.Comellas 2026):
-
- Z_13 x(805) Z_1563, generators [5,451]<->[8,1232]: [10,746]<->[3,118]: [6,537]<->[7,90]: . Avg. dist.:5.996850 (1, 6, 30, 150, 742, 3440, 10490, 5460) transm.: 121844 adjlist NX
-
- Z_39 x(101) Z_521, generators [22,223]<->[17,306]: [21,277]<->[18,385]: [11,315]<->[28,213]:. Avg. dist.:5.996850 (1, 6, 30, 150, 742, 3440, 10490, 5460) transm.: 121844 . adjlist NX
Former result, Order = 19383
Communicated by Eyal Loz, Math Dep., Auckland Univ., New Zealand (July 2006)
Voltage graph Z_21 x(48) Z_923, B(0,3), voltages [(19,865)(7,330)(11,97)], avg. dist.: 5.979465
Download the adjlist NX version of the graph. Link to Eyal Loz's original data.
Communicated July 2006.
Found also from other semidirect products (F.Comellas 2024):
Z_21 x(250) Z_923 with generators [2,620 ]<>[19,324 ]:[11,486 ]<>[10,401 ]:[7,279 ]<>[14,76 ]. Avg.dist.: 5.979465 (1,6,30,150,734,3386,10020,5056) transm.: 115894 adjlist NX
Z_21 x(529) Z_923 with generators [13,81 ]:[14,353 ]<>[7,73 ]:[19,422 ]<>[2,333 ]. Avg.dist.: 5.979465 (1,6,30,150,734,3386,10020,5056) transm.: 115894 adjlist NX
Rajiv_53456
Degree= 7, Diameter = 7; Order =53456; Moore bound= 391909
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 Z16 (⋊8) Z3341, with generators [1,1477 ]<>[15,233 ]:[6,1030 ]<>[10,1927 ]:[10,1927 ]<>[6,1030 ]:[15,233 ]<>[1,1477 ].
Avg.dist: 6.0247867 . Dist. distrib. :[1, 7, 42, 252, 1511, 8455, 29421,13767 ]
. 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= 53020.
Cayley graph. Found as a semidirect product, 53020 nodes and 185570 edges (F.Comellas 2024):
** Z_20 x(729) Z_2651, generators [6,894 ]<>[14,1583 ]:[17,2271 ]<>[3,140 ]:[18,2411 ]<>[2,928 ]:[10,1210 ]<>[10,1210 ]. Avg. dist.: 6.033799 (1,7,42,252,1500,8326,28815,14077) transm.: 319906. adjlist NX
** Z_20 x(970) Z_2651, generators [1,533]<>[19,2522]:[18,2238]<>[2,167]:[3,1599]<>[17,170]:[10,1661]<>[10,1661]. Avg. dist.: 6.025896 (1,7,42,252,1510,8414,29028,13766) transm.: 319487. adjlist NX
Former result, Order = 52768
Communicated by Eyal Loz, Math Dep., Auckland Univ., New Zealand (July 2006)
Voltage graph Z_32 x(19) Z_1649, B(1,3), voltages [31,96 ]:[24,586]:[21,1521]:[16,1088], avg. dist.: 6.038661 (1,7,42,252,1504,8357,28241,14364) transm.: 318642
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.
Dhar_135621
Degree= 8, Diameter = 7; Order =135621; Moore bound = 1098056
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 from the semidirect group Z27 (⋊88) Z5023. Generators:[13, 3021]<> [14, 1419] : [21, 1456] <> [6, 1760] : [16, 3771] <> [11, 1610] : [17, 329] <> [10, 83]; avg. dist. 6.1133305 ; dist. distr. [1, 8, 56, 392, 2744, 18016, 74084, 40320]
Former result, order=133887.
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 Z39 (⋊71) Z3433, with generators [1,2494 ]<>[38,2866 ]:[5,1376 ]<>[34,3235 ]:[22,2112 ]<>[17,1712 ]:[37,1255 ]<>[2,564 ]
Avg.dist: 6.112064 . Dist. distrib. :[1, 8, 56, 392, 2736, 17820, 73132, 39742 ]
. In this link
you can download the adjacency list, a verifier (standard-library Python), and a paper describing the construction of the graph.
Download here the raw adjacency list .
Dhar_286770
Degree= 9, Diameter = 7; Order =286770; Moore bound = 2696338
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 from the semidirect group Z30 (⋊766) Z9559. Generators:
[2,1000]<>[28,1406]:[4,2374]<>[26,8203]:[20,3753]<>[10,6411]:[27,5314]<>[3,8985]:[15,553]<>[15,553]
Rajiv_589734
Degree= 10, Diameter = 7; Order =589734; Moore bound = 5978710
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 Z27 (⋊1369) Z21842, with generators [7,13766 ]<>[20,3878 ]:[10,12156 ]<>[17,16280 ]:[13,21389 ]<>[14,6711 ]:[16,1327 ]<>[11,3161 ]:[19,18286 ]<>[8,16198 ]:
Avg.dist: 6.1849562684. Dist. distrib: [ 1, 10 ,90, 810, 7250, 61908, 331336, 188329 ]
. In this link
you can download the adjacency list, a verifier (standard-library Python), and a paper describing the construction of the graph.
Dhar_1031240
Degree= 11, Diameter = 7; Order =1031240; Moore bound = 12222222
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 from the semidirect groupZ14 (⋊111) Z73660 Generators:
[1,599]<>[13,72991]:[6,46276]<>[8,31744]:[9,48194]<>[5,65626]:[12,43845]<>[2,8195]:[11,34235]<>[3,26815]:[7,15080]<>[7,15080]. Avg.dist: 6.19087797. Dist. distrib: [ 1, 11, 110, 1100, 10952, 102859, 590802, 325405]
Rajiv_2006692
Degree= 12, Diameter = 7; Order =2006692 ; Moore bound = 23384604
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 Z31 (⋊1905) Z64732, with generators [3,6544 ]<>[28,36800 ]:[7,61265 ]<>[24,45463 ]:[13,63227 ]<>[18,42325 ]:[16,24277 ]<>[15,48071 ]:[20,2592 ]<>[11,38792 ]:[25,36675 ]<>[6,50469 ].
Avg.dist: 6.24260673 . Dist. distrib: [ 1, 12, 132, 1452, 15956, 168136, 1129168, 691835].
. In this link
you can download the adjacency list, a verifier (standard-library Python), and a paper describing the construction of the graph.
Results for diameter 7 and degrees 13 and 15 obtained by Eyal Loz and Jozef Širáň. New record graphs in the degree-diameter problem.
Australas. J. Combin. 41 (2006), 63–80..
K1Σ8 H11
Degree = 14, Diameter = 7; Order =6200460; Moore bound= 73206603 ;
J. Gómez, M.A. Fiol and O. Serra, On large (Δ, D) graphs,
Discrete Mathematics, 114 (1993), pp. 235. link to the paper
K1Σ8 H13
Degree = 16, Diameter = 7; Order =14882658; Moore bound= 195267857 ;
J. Gómez, M.A. Fiol and O. Serra, On large (Δ, D) graphs,
Discrete Mathematics, 114 (1993), pp. 235. link to the paper