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

Largest Known (Degree, Diameter)-Graphs

Diameter 10

Last modification: October 14, 2026.

http://www-mat.upc.es/grup_de_grafs/desc_g10.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]].

Conder_1250

Degree = 3, Diameter = 10; Order =1 250; Moore bound=3070;

Communicated by Marston Conder ( m.conder@auckland.ac.nz ) on August 17, 2006. http://www.math.auckland.ac.nz/~conder/symmcubic2048list.txt

Marston Conder, Jicheng Ma. Arc-transitive abelian regular covers of cubic graphs. J. Algebra, 387 (2013) 215-242.

Download the raw adjacency list of the graph.

Download the adjacency list (NetworkX) 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 .

Dahr_18436

Degree= 4, Diameter = 10; Order =18436; Moore bound=118097.

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 4, diameter 10, and 18436 vertices is constructed as a two-copy voltage lift over the semidirect product Z₁₁ ⋉₇₅₃ Z₈₃₈, of order 9218. The group operation is defined by [x,y] × [u,v] = [(x+u) mod 11, (y·753^u+v) mod 838]. The construction uses two copies of the group, with inverse-pair generators [5,467] and [10,637] for the first and second copies, respectively, and two generators [8,302] and [10,5] connecting the two copies. Each connecting generator also defines an inverse edge in the opposite direction. The resulting graph has 2·9218 = 18436 vertices.

Download the local edge 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 = 18369

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_157 (x9) Z_397. Order: 18369 / Size: 36738 / 4-reg.? True / Girth: 12 / Diam.: 10 / Avg.dist: 8.17560 / Aut.group.ord.: 6123 / Automorphism group structure C157 x| C39

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 = 17703

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.

Former result, order =17604

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

Obtained by Sampels (Sampels@Informatik.Uni-Oldenburg.DE) (july 1997). Link to generators.

Dahr_190578

Degree= 4, Diameter = 10; Order =190578; Moore bound=.

Voltage lift of a base graph with 2 vertices over the metacyclic group G = Z_1381 x|_397 Z_69 of order 95289: the vertices are the pairs (copy, group element) and (c,x) is adjacent to (c',x g) for each of the 5 units (loop pairs inside a copy, edges between copies) listed in the json.

Download this file with a .json file and other details to reconstruct the graph.

Dhar_1279794

Degree= 6, Diameter =10; Order = 1279794; Moore bound = 14648436

A graph constructed as a two-copy voltage lift over the semidirect product Z₅₁⋉₇₆₄₃Z₁₂₅₄₇, of order 639897, with a group operation defined as in the original contributions from M.J. Dinneen and P. Hafner (see top of this page) The construction uses two copies of the group, with inverse-pair generators [16,7291] and [32,1734] for the first and second copies, respectively, corresponding to the integers 208043 and 403238. The copies are connected by four generators [17,374], [16,6429], [6,11155], and [39,99], corresponding to the integers 213673, 207181, 86437, and 489432, respectively. Each connecting generator also defines its inverse edge in the opposite direction. The resulting graph has 2·639897 = 1279794 vertices.

Download the compressed local raw adjacency list of the graph.

Former result, order = 1254594.

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 = 1253751.

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 Z₄₃ (⋊₄) Z₂₉₁₅₇, with generators Avg.dist: 8.61308186. Dist. distrib. : [ 1, 6, 30, 150, 750, 3750, 18622, 89010, 355526, 661676, 124230].

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

Current entries for diameter 10 and degrees 7 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)

Entries for diameter 10 and degrees 8,12, 14 and 16 found by Dharunish Yugeswardeenoo (dharyugi@gmail.com; August 2026). Result for diameter 10 and degree 9 found by Chenxudong, Hangzhou, China (qq475519905@gmail.com; September 2026). Results for diameter 10 and degrees 5, 7, 11, 13 and 15 obtained by Eyal Loz and Jozef Širáň details at New record graphs in the degree-diameter problem. Australas. J. Combin. 41 (2008), 63-АУ80. .