Table of Large Degree/Diameter Graphs
Description
A graph, G=(V,E), consists of a non empty finite set V of elements called vertices and a set E of pairs of elements of V called edges. The number of vertices N=|G|=|V| is the order of the graph. If (x,y) is an edge of E, we say that x and y (or y and x) are adjacent and this is usually written x --> y. It is also said that x and y are the endvertices of the edge (x,y). The degree of a vertex δ(x) is the number of vertices adjacent to x. The degree of G is Δ=max_{x ∈ V} δ(x). A graph is regular of degree Δ or Δ - regular if the degree of all vertices equal Δ. The distance between two vertices x and y, d(x,y) , is the number of edges of a shortest path between x and y , and its maximum value over all pair of vertices, D=max_{x, y ∈ V}d(x,y) , is the diameter of the graph. A (Δ,D) graph is a graph with maximum degree Δ and diameter at most D. The order of a graph with degree Δ, Δ > 2), of diameter D is easily seen to be bounded by 1 + Δ + Δ (Δ-1) + ...+ Δ (Δ-1) D-1 = (Δ (Δ-1)D -2) / (Δ-2) = N(Δ, D) Hoffman and Singleton introduced the concept of Moore graphs, after Edward Forrest Moore, as graphs attaining this value, known as Moore bound. They also showed that, for D ≥ 2 and Δ ≥ 3, Moore graphs exist for D=2 and Δ =3,7 , and (perhaps) 57. In this context, it is of great interest to find graphs which for a given maximum diameter and maximum degree have a number of vertices as close as possible to the Moore bound. Download the package, unpack it and open in a browser the file table_degree_diameter.html or the file taula_delta_d.html. The table on that page presents the state of the art as of June 2025 for the largest known (Δ, D)-graphs. Entries in boldface are optimal. Click on a position to view more information about that entry, including graph construction details, the Moore bound, author, references, and more. Entries with a border include a SageMath script to compute their relevant properties. Adjacency lists are available for most graphs with fewer than 20,000 vertices. By clicking on entry (8,3) = 253, you can access a ZIP file containing the programs used to obtain the results for this graph, as well as for the graphs (3,5), (6,8), (7,6), (7,7), (8,5), (9,4), (10,4), (10,5), (11,5), (12,5), (13,5), (14,5), and (15,5) -- all found by the author in 2024. The C program used is the same as the one that found the entry (8,3) in 1994, with minor modifications to the output. Journal publications associated with this data: F. Comellas. Table of large graphs with given degree and diameter. arXiv:2406.18994 [math.CO]. doi: 10.48550/arXiv.2406.18994 F. Comellas. New results on the degree-diameter problem for undirected graphs. Electron. J. Graph Theory Appl. 13 (1) (2025), 211-215. doi:10.5614/ejgta.2025.13.1.14.
Files
Steps to reproduce
Download the package delta-d@250522.zip, unzip it and open in a browser the file table_degree_diameter.html or the file taula_delta_d.html. Click on a position to view more information about that entry, including graph construction details, the Moore bound, author, references, and more. Entries with a border include a SageMath script to compute their relevant properties. Adjacency lists are available for most graphs with fewer than 20,000 vertices. By clicking on entry (8,3) = 253, you can access a ZIP file containing the programs used to obtain the results for this graph, as well as for the graphs (3,5), (6,8), (7,6), (7,7), (8,5), (9,4), (10,4), (10,5), (11,5), (12,5), (13,5), (14,5), and (15,5) -- all found by the author in 2024. The C program used is the same as the one that found the entry (8,3) in 1994, with minor modifications to the output.