Large graphs of given maximum degree and diameter (Degree-Diameter Problem): 34 constructions

Dharunish Yugeswardeenoo, 25 October 2026

Thirty-four graphs for the degree-diameter problem. Each has maximum degree Delta and diameter D, and on 25 October 2026 each was a largest known graph for its pair (Delta, D): (4,7) 1,344; (6,6) 8,082; (7,5) 2,810; (8,6) 40,560; (8,7) 138,138; (8,8) 779,817; (8,9) 4,478,321; (8,10) 26,899,143; (9,6) 77,418; (9,7) 304,038; (10,4) 2,485; (10,6) 137,180; (10,7) 616,980; (10,8) 4,513,497; (10,9) 34,219,967; (11,2) 106; (11,4) 3,444; (11,5) 37,628; (11,6) 186,033; (12,3) 810; (12,5) 42,342; (12,6) 364,032; (12,7) 2,115,075; (12,9) 178,084,211; (13,3) 972; (13,5) 51,417; (13,6) 538,740; (14,4) 8,443; (14,6) 824,124; (14,8) 64,439,767; (15,2) 190; (15,4) 13,608; (15,6) 1,428,361; (16,6) 1,793,786 Version 2 adds a larger graph for four of the pairs -- (13,6) 538,440 to 538,740, (14,4) 8,442 to 8,443, (15,6) 1,426,903 to 1,428,361 and (16,6) 1,790,870 to 1,793,786 -- and keeps the superseded graphs in the file list; CURRENT_GRAPHS.txt names the current file for each pair. Twenty-two graphs are given as adjacency lists and twelve as Cayley-graph or voltage-lift data over metacyclic groups. Two C programs check the maximum degree and the diameter by breadth-first search from every vertex. README.md lists the graphs, the file formats, the checks,

Dharunish Yugeswardeenoo, 24 October 2026


Thirty-two graphs for the degree–diameter problem. Each has maximum degree Δ and diameter D, and on 24 October 2026 each was the largest known graph for its pair (Δ, D): (4,7) 1,344; (6,6) 8,082; (7,5) 2,810; (8,6) 40,560; (8,7) 138,138; (8,8) 779,817; (8,9) 4,478,321; (8,10) 26,899,143; (9,6) 77,418; (9,7) 304,038; (10,6) 137,180; (10,7) 616,980; (10,8) 4,513,497; (10,9) 34,219,967; (11,4) 3,444; (11,5) 37,628; (11,6) 186,033; (12,3) 810; (12,5) 42,342; (12,6) 364,032; (12,7) 2,115,075; (12,9) 178,084,211; (13,3) 972; (13,5) 51,417; (13,6) 538,440; (14,4) 8,442; (14,6) 824,124; (14,8) 64,439,767; (15,2) 190; (15,4) 13,608; (15,6) 1,426,903; (16,6) 1,790,870. Twenty graphs are given as adjacency lists and twelve as Cayley-graph or voltage-lift data over metacyclic groups. Each file gives a graph of maximum degree Δ and diameter D (the degree–diameter problem). Two C programs check the maximum degree and the diameter by breadth-first search.  
On 24 October 2026 each was the largest known graph for its pair (Δ, D); compare the table maintained by F. Comellas, https://web.mat.upc.edu/francesc.comellas/delta-d/table_degree_diameter.html.

Constructions

(4,7), 1,344. Lift of a two-vertex base graph over SL(2,7):C2, which has order 672 and is SmallGroup(672,1047) in GAP. Each base vertex carries two semi-edges whose voltages are involutions. The two base vertices are joined by a double edge whose voltages have orders 8 and 6. The six voltages generate the group. Vertex (c, g) is joined to (c, gW) for each semi-edge voltage W at c, and (0, g) is joined to (1, gW) for each of the two edge voltages W. The lift is 4-regular on 2 × 672 vertices.

(6,6), 8,082. Start from the incidence graph of the split Cayley hexagon H(5) (7,812 vertices, 6-regular, diameter 6) and a Hermitian spread of 126 pairwise opposite lines. On 30 spread lines 3 of the incident edges are subdivided, and on 90 spread lines 2 of them (270 new vertices). On each such line, exactly the points whose cyclic label lies in a fixed difference cover of Z_6 keep their edges. The new vertices at one spread line form a clique, and their remaining degree is used by external edges. These are invariant under a subgroup of order 10 of SL(2,5), which acts regularly on the 120 subdivided spread lines.

(8,6), 40,560; (9,6), 77,418; (10,6), 137,180; (14,6), 824,124. Start from the incidence graph of H(q), for q = 7, 8, 9 and 13, which is (q+1)-regular of diameter 6, and a Hermitian spread of q³+1 pairwise opposite lines. The q+1 lines of one regulus are unchanged. On each of the other q³−q spread lines, k of its q+1 incident edges are subdivided, with k = 4, 5, 6 and 9 respectively. The points of each such line carry cyclic labels in Z_{q+1}, given by the hyperplanes through the 3-space of the fixed regulus. Exactly the points whose label lies in a difference cover of Z_{q+1} keep their edges. The new vertices of one spread line form a clique, and each new vertex has q − k further external edges (3, 3, 3 and 4). The external edges are invariant under a group isomorphic to SL(2,q), which acts regularly on the q³−q subdivided spread lines and preserves the labels.

(12,6), 364,032. Start from the incidence graph of H(11) (354,312 vertices, 12-regular, diameter 6) and a Hermitian spread of 1,332 pairwise opposite lines. On 480 spread lines 8 of the incident edges are subdivided, and on 840 spread lines 7 of them (9,720 new vertices). On each such line, exactly the points whose cyclic label lies in a fixed difference cover of Z_12 keep their edges. The new vertices at one spread line form a clique, and their remaining degree is used by external edges. These are invariant under a subgroup of order 120 of SL(2,11), which acts regularly on the 1,320 subdivided spread lines.

(7,5), 2,810, and the Cayley graphs (8,7), (8,8), (8,9), (8,10), (9,7), (10,7), (10,8), (10,9), (12,7), (12,9), (14,8). These are lifts over the metacyclic groups in the table, fully specified by their lift files. (7,5) has two copies, with a loop pair at each copy and five edges between the copies. The others are Cayley graphs.

(11,4), 3,444. Start from the incidence graph of the symplectic generalized quadrangle W(9) (820 points, 820 lines, 10-regular, diameter 4, girth 8). Every vertex a gets two copies (a, 0) and (a, 1). A base edge a–b carries a voltage x ∈ {0, 1} and joins (a, i) to (b, i + x mod 2). For each of the 597,780 pairs of opposite vertices, the 10 geodesics between them do not all have the same voltage sum. The two copies of every line are joined to each other. For each of the 82 lines of a spread and each sheet i, a new vertex is joined to the 10 point copies adjacent to the copy of that line in sheet i, and the two new vertices of a spread line are adjacent. The result has 3,280 + 164 = 3,444 vertices and is 11-regular.

(11,5), 37,628. A partial fold of H(9), built 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 37,622 vertices. Six further vertices are added, each joined to vertices with a free degree slot (50 new edges). No vertex or edge of the 37,622-vertex graph is removed.

(12,5), 42,342. A partial fold of H(9), built on the construction and duality data of R. Rajiv: a closed split set, copies of H(3) and pencil vertices. A further 165 vertices are added, each joined to at most 12 vertices with a free degree slot, so that the balls of radius 4 around their neighbours cover the graph.

(13,5), 51,417. A partial fold of H(9) on R. Rajiv's (13,5) split set. The pencil vertices carry extra anchors, and twin vertices fill the free slots that remain.

(11,6), 186,033. R. Rajiv's (11,6) graph of order 182,618 with 3,415 added vertices. Each added vertex is joined to vertices that had a free degree slot, and in some cases to other added vertices. No edge of Rajiv's graph is changed.

(12,3), 810. Lift of a 3-vertex base multigraph. The vertices are triples (i, z, x) with i a base vertex, z ∈ Z_10 and x ∈ F_3^3, so the order is 3 × 10 × 27. A base dart i → j carries a ∈ Z_10 and an affine map (M, u) of F_3^3, and joins (i, z, x) to (j, z + a, Mx + u). The graph is 12-regular.

(13,3), 972. The controller C is the voltage lift of a 3-vertex base graph over Z_12. Its base edges are 0–1 with voltages 0 and 3, 0–2 with voltages 3 and 4, and 1–2 with voltages 6 and 11, so C is 4-regular on 36 vertices with diameter 3. Every vertex of C becomes a block of 27 points, GF(3)^3. Every edge a → b of C carries an affine relation: point x of block a is adjacent to the three points Mx + c + λw of block b (λ ∈ GF(3), M invertible, w ≠ 0). Every block also carries an involution x ↦ Px + t (P² = I, Pt = −t), which adds one edge at each point it moves. The graph has 972 vertices and 6,264 edges: 864 vertices of degree 13 and 108 of degree 12.

(13,6), 538,440. Built on R. Rajiv's compound graph H_9(T_4): four blocks, each a copy of the incidence graph of H(9) (realised on the parabolic quadric Q(6,9)), joined by a tournament of point–line matchings and by point–point and line–line matchings. As in Rajiv's surgery, the type-preserving matchings are cut at a set S_P of points and a set S_L of lines. Here S_P is clustered (every cut point has all its cut partners on one line), and S_L is clustered dually. S_P is the 7,300 points of a Hermitian spread: the 730 lines of H(9) in a hyperplane meeting Q(6,9) in an elliptic quadric Q⁻(5,9). S_L is the 7,300 lines through the 730 points of a distance-3 ovoid. 7,000 new vertices are added, each joined to one cut point and one cut line of every block, and they are joined among themselves by a 5-regular graph Y. A further 7,067 edges join vertices with free degree. The order is 4 × 2 × 66,430 + 7,000.

(15,6), 1,426,903. The same surgery on R. Rajiv's compound graph H_11(T_4), with four blocks, each a copy of the incidence graph of H(11). S_P is the 15,984 points of a Hermitian spread (1,332 pairwise opposite lines in an elliptic hyperplane section). S_L is a clustered set of 9,655 lines: 86 orbits of length 111 under a cyclic automorphism of H(11) of order 111, plus 109 further lines. 9,655 new vertices are added, each joined to one cut point and one cut line of every block, and they are joined among themselves by a 7-regular graph Y. Ten further edges are added. The order is 4 × 2 × 177,156 + 9,655.

(16,6), 1,790,870. The same surgery on R. Rajiv's compound graph H_11(T_5), with five blocks joined by a regular tournament, using the same S_P (15,984 points) and S_L (9,655 lines). Every cut element frees two slots in every block. 19,310 = 2 × 9,655 new vertices are added, each joined to one cut point and one cut line of every block, and they are joined among themselves by a 6-regular graph Y. A further 3,847 edges are added. The order is 5 × 2 × 177,156 + 19,310.

(14,4), 8,442. F. Comellas' (14,4) graph of order 8,434 (2026; the file deg14D4_8434_.edges on the table site, sha256 ddd4daae7a661ed870aaa91f37da76f6bb3346821e6c284d47fe3c9a1ac56e60) with 8 added vertices, 8,434 to 8,441, each of degree 14. 286 edges of the base graph are removed and 134 are added: the 112 edges at the new vertices and 22 others.

(15,2), 190. An explicit graph on 190 vertices, given by its adjacency list.

(15,4), 13,608. A polarity quotient of a bipartite route-chart lift. The group is G = SmallGroup(336,20) ≅ C7 ⋊ (C3 × QD16). It has an index-2 subgroup, and the odd elements form the other coset. The letters are a^{±1} and b^{±1} (odd elements of order greater than 2) and c (an odd involution). The lift B has vertices (h, x) with h ∈ G and x ∈ F_3^4. Its edges are (h, x) ~ (ht, A_t x + λ b_t) for every letter t and every λ ∈ F_3, where A_{t⁻¹} = A_t⁻¹, b_{t⁻¹} = A_t⁻¹ b_t, A_c² = I and A_c b_c = b_c. B is a 15-regular bipartite graph on 336 × 81 = 27,216 vertices with diameter 5. The map ρ(h, x) = (c₀h, x), with c₀ an odd involution, is a fixed-point-free automorphism that exchanges the two sides. The quotient B/ρ has vertices {u, ρu}, with [u] ~ [w] if and only if u ~ w. It has 13,608 vertices and diameter 4; 54 absolute vertices have degree 14 and the rest degree 15.