Largest Known (Degree, Diameter)-Graphs
Diameter 10
Last modification: October 15, 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: Zm ⋉a Zn,
of order m·n. Group operation defined as [x,y] × [u,v] = [(x+u) mod m, (y·au+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 .
Bhakar_19306
Degree= 4, Diameter = 10; Order =19306; Moore bound=118097.
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 graph is constructed as a two-copy voltage lift over the semidirect
product Z49 ⋉16 Z197, of order 9653.
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 49, (y·16u+v) mod 197].
The construction uses two copies of the group, labelled 0 and 1, with internal inverse-pair generators
(48,0) and (39,177), respectively, corresponding to the integers 9456 and 7860.
The copies are connected by two generators: (0,0) and (37,28), corresponding to the integers 0 and 7317, respectively. Each connecting generator also defines its inverse edge in the opposite direction.
The resulting graph has 2·9653 = 19306 vertices, is 4-regular, and has diameter 10.
The construction was obtained through an exhaustive symmetry-reduced search over two-vertex voltage lifts
of odd-order metacyclic groups.
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 graph including symmetry group sizes and the number of k-cycles (k=3..). This is the online version .
Former result, order=18436
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·753u+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.
Bhakar_202550
Degree= 5, Diameter = 10; Order =202550; Moore bound= 1747626
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.
The graph is constructed as a two-copy voltage lift over the semidirect
product Z25 ⋉4Z4051, of order 101275.
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 25, (y·4u+v) mod 4051].
The construction uses two copies of the group, labelled 0 and 1, with internal inverse-pair generators
(22,0) and (17,2037), respectively, corresponding to the integers 89122 and 70904.
The copies are connected by three generators: (0,0), (23,1266), and (10,1706), corresponding
to the integers 0, 94439, and 42216, respectively.
Each connecting generator also defines its inverse edge in the opposite direction.
The resulting graph has 2·101275 = 202550 vertices, is 5-regular, and has diameter 10.
The construction was obtained through a systematic search of two-vertex voltage lifts over odd-order
metacyclic groups.
In this link you can download the edge list and more information detailing the construction of the graph.
Download the local compressed edge list of the graph.
Former result, order=190578
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.
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.
Entry for diameter 10 and degree 8 found by Dharunish Yugeswardeenoo (dharyugi@gmail.com; Chalfont, PA, USA; October 19, 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.
The graphs are Cayley graphs of metacyclic groups arising from semidirect products of cyclic groups..
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).
Current entries for diameter 10 and degrees 6 to 16, except 8, 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. All are Cayley graph constructed over semidirect products.
In this link you can download more information detailing the construction of 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 6 to 16 found by Dharunish Yugeswardeenoo (dharyugi@gmail.com; Chalfont, PA, USA; October 6-14, 2026). See above.
Result for diameter 10 and degree 9 found by Chenxudong, Hangzhou, China (qq475519905@gmail.com; September 2026).
Entries for diameter 10 and degrees 8,12, 14 and 16 found by Dharunish Yugeswardeenoo (dharyugi@gmail.com; August 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--€“80. .