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

Largest Known (Degree, Diameter)-Graphs

Diameter 7

Last modification: October 15, 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.

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]].

Bakhar_200

Degree = 3, Diameter = 7; Order =200; Moore bound=382;

Obtained by Nishant Bhakar (nishant.bhakar@gmail.com, Redmond, WA, USA; October 9, 2026). Found with Claude Fable assisting with implementation and the analysis and summarization of the results.

Graph constructed as a regular voltage lift of a five-vertex base graph over the semidirect product Z₈ ⋉₄ Z₅, of order 40. The group operation (following the notation of the original contributions to this table -see top of this page-) is defined by [a,b] × [u,v] = [(a+u) mod 8, (b·4^u+v) mod 5]. In a voltage lift, each base vertex v is replaced by 40 vertices (v,g), one for each group element g in G. A directed base edge v → w with voltage t produces the edges (v,g)–(w,gt) for all g in G. The base graph has five vertices, labelled 0, 1, 2, 3, and 4, and five directed links: 0 → 1, 0 → 2, 0 → 3, and 1 → 4, all with identity voltage (0,0), and 1 → 2 with voltage (7,0). In addition, vertex 2 has a semi-edge with involution voltage (4,0), connecting (2,g) to (2,gt), and contributing one neighbor per lifted vertex. Vertex 3 has a loop with voltage (6,1), and vertex 4 has a loop with voltage (6,2); each loop contributes the two edges corresponding to its voltage and its inverse. The resulting graph has 5·40 = 200 vertices, 300 edges,is 3-regular, and has diameter 7. The construction was obtained through an exhaustive search of voltage assignments for five-vertex cubic base graphs over groups of order 40.

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 =196

Communicated by G. Exoo (January 28, 2008). G.E. data

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

Dhar_1344

Degree= 4, Diameter = 7; Order =1344; Moore bound=4373.

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.

This graph is constructed as a regular voltage lift of a two-vertex base graph over the group G = SL(2,7) : C₂, of order 672 (SmallGroup(672,1047) in GAP). In this voltage lift, each base vertex c is replaced by 672 vertices (c,g), one for each element g of G, and a directed edge from c to d with voltage W produces the edges (c,g)–(d,gW) for all g in G. The base graph has two vertices, labelled 0 and 1, each carrying two semi-edges, and the vertices are joined by two parallel edges. The semi-edges at vertex 0 have voltages W1 and W2, while those at vertex 1 have voltages W3 and W4. These four voltages are involutions, each contributing one neighbor to every lifted vertex. The two parallel edges from 0 to 1 have voltages W5 and W6, of orders 8 and 6, respectively, and their reverse edges carry the inverse voltages. The six voltages, given explicitly as permutations of 32 points, generate the entire group G. Each voltage below is a permutation of {1..32}, written as its 32 images:

W1 32 11 22 28 13 12 19 29 30 31 2 6 5 16 23 14 21 26 7 24 17 3 15 20 27 18 25 4 8 9 10 1

W2 17 30 18 25 14 16 23 10 19 8 15 20 24 5 11 6 1 3 9 12 28 31 7 13 4 29 32 21 26 2 22 27

W3 28 20 29 32 17 9 14 22 6 19 18 25 15 7 13 26 5 11 10 2 30 8 27 31 12 16 23 1 3 21 24 4

W4 14 31 15 20 28 21 26 9 8 23 29 32 27 1 3 25 24 30 22 4 6 19 10 17 16 7 13 5 11 18 2 12

W5 25 26 10 17 3 27 31 18 23 28 4 8 21 24 30 13 29 32 1 16 14 11 2 6 22 15 20 12 19 7 9 5

W6 24 1 30 16 9 3 4 23 22 18 14 15 8 28 29 2 25 10 20 32 27 12 6 7 17 31 21 19 5 13 26 11

Each lifted vertex has two neighbors within its own copy and two neighbors in the other copy, giving a 4-regular graph with 2·672 = 1344 vertices and 2688 edges.

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=1320

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 )

Bhakar_6000

Degree= 5, Diameter = 7; Order =6000; Moore bound=27 306.

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 non-metacyclic group G = ((F₅² ⋊ C₃) × Z₅) ⋊ C₈, of order 3000. The elements of F₅² are represented by pairs (a,b), and the generator c of order 3 acts by the matrix B = [[0,4],[1,4]], while the generator y of order 8 acts by the matrix A = [[1,1],[2,4]] on F₅², and sends c to c² and z to z², where z has order 5. The construction uses two copies of G, with inverse-pair generators y and ((0,2),c,z⁴)y⁴ for the first and second copies, respectively. The copies are connected by three generators: the identity 1, ((0,1),c,z)y⁴, and ((1,4),c,z⁴)y⁷. Each connecting generator also defines its inverse edge in the opposite direction. The resulting graph has 2·3000 = 6000 vertices, is 5-regular, and has diameter 7.

In this link you can download the edge list and more information detailing the construction of the graph.

Download the local raw adjacency list of the graph.

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

Former result, order=5896

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 5, diameter 7, and 5896 vertices is constructed as a two-copy voltage lift over the semidirect product Z₁₁ ⋉₂₆₅Z₂₆₈, of order 2948. The group operation is defined by [x,y] × [u,v] = [(x+u) mod 11, (y·265^u+v) mod 268]. The construction uses two copies of the group, with inverse-pair generators [2,142] and [4,182] for the first and second copies, respectively, and three generators [3,164], [0,74], and [8,165] connecting the two copies. The resulting graph has 2·2948 = 5896 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=5894

Obtained by Michel de Messieres (michel@demessieres.com, October 6, 2026). The search programs were written and run by an AI assistant (Anthropic's Claude) under the author's direction. This graph is a lift of a 2-vertex voltage graph over Z₇ (⋊₃₃) Z₄₂₁. The graph is vertex-transitive and edge-transitive but not Cayley.

The graph and a detailed write-up are also on Zenodo https://doi.org/10.5281/zenodo.23176213

Download the edge list of the graph.

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

Former result, Order = 5824

Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, Septembert 30, 2026) with the use of AI agentic coding systems Anthropic's Claude Fable 5.1 and Opus 5.5. Orbit graph. G = Sz(8), of order 29120, with one orbit, of size 5824 (a point stabilizer has order 5). This graph is vertex-transitive and edge-transitive

In this link: https://doi.org/10.5281/zenodo.23057003 you can download the details and a paper describing the construction of the graph.

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 = 5516

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 ) .

Rithwik_20400

Degree= 6, Diameter = 7; Order =20400; Moore bound= 117186

Rithwik Gupta (rithwikca2020@gmail.com ); September 8, 2026). Cayley graph found through AI-assisted exhaustive computational searches (details to come).

Download the local edge list .

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

Former result, Order = 20319

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_56952

Degree= 7, Diameter = 7; Order =56952; Moore bound= 3919010

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 semidirect product Z₂₄ (⋊₄₄) Z₂₃₇₃ S= {3082,5577,17883,29904,40986,53484,55264}.

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= 53456.

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.

Current entries for diameter 7 and degrees 8, 10 and 12, found by Dharunish Yugeswardeenoo (dharyugi@gmail.com; Chalfont, PA, USA; October 6-19, 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. All three graphs are Cayley graphs of metacyclic groups arising from semidirect products of cyclic groups. 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).

Current entries for diameter 7 and degrees 9,11,13, 15 and 16, 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 graphs 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

Entries for diameter 7 and degrees 8 to 15 , except degree 14, found by Dharunish Yugeswardeenoo (dharyugi@gmail.com; Chalfont, PA, USA; October 6-14, 2026). See above

Results for diameter 7 and degrees 13 and 15 obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca, Septembert 30, 2026). In this link: https://doi.org/10.5281/zenodo.23057003 you can download the details and a paper describing the construction of the graphs.

Results for diameter 7 and degrees 4 to 16 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

Bhakar _14995367

Degree = 16, Diameter = 7; Order =14995367; Moore bound= 195267857 ;

Cayley graph with degree 16, diameter 7, and 14995367 vertices constructed over the semidirect product Z₁₇₉ ⋉₇₂₂₁₁ Z₈₃₇₇₃, of order 14995367 (with 83773 prime). 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 179, (y·72211^u+v) mod 83773]. The graph has eight inverse-pair generators (164,42288), (163,71803), (121,66096), (146,27388), (167,16009), (110,14308), (108,25621), and (172,49013), corresponding to the integers 13781060, 13726802, 10202629, 12258246, 14006100, 9229338, 9073105, and 14457969, respectively. The graph was obtained using a coincidence-free level pattern followed by simulated annealing with targeted moves.

In this link you can download the edge list and more information detailing the construction of the graph.

Download the local raw adjacency list of the graph.

Former result, Order = 14882658

K₁Σ₈ H₁₃. J. Gómez, M.A. Fiol and O. Serra, On large (Δ, D) graphs, Discrete Mathematics, 114 (1993), pp. 235. link to the paper