In mathematics, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and i = d(v, w).
Every distance-transitive graph is distance-regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group.
It turns out that a graph of diameter is distance-regular if and only if there is an array of integers such that for all , gives the number of neighbours of at distance from and gives the number of neighbours of at distance from for any pair of vertices and at distance on . The array of integers characterizing a distance-regular graph is known as its intersection array.
Suppose is a connected distance-regular graph of valency with intersection array . For all : let denote the -regular graph with adjacency matrix formed by relating pairs of vertices on at distance , and let denote the number of neighbours of at distance from for any pair of vertices and at distance on .
- for all .
- and .
- for any eigenvalue multiplicity of , unless is a complete multipartite graph.
- for any eigenvalue multiplicity of , unless is a cycle graph or a complete multipartite graph.
- if is a simple eigenvalue of .
- has distinct eigenvalues.
If is strongly regular, then and .
Some first examples of distance-regular graphs include:
Classification of distance-regular graphs
Cubic distance-regular graphs
The cubic distance-regular graphs have been completely classified.
The 13 distinct cubic distance-regular graphs are K4 (or tetrahedron), K3,3, the Petersen graph, the cube, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the dodecahedron, the Desargues graph, Tutte 12-cage, the Biggs–Smith graph, and the Foster graph.
- Bang, S.; Dubickas, A.; Koolen, J. H.; Moulton, V. (2015-01-10). "There are only finitely many distance-regular graphs of fixed valency greater than two". Advances in Mathematics. 269 (Supplement C): 1–55. arXiv:0909.5253. doi:10.1016/j.aim.2014.09.025.
- Godsil, C. D. (1988-12-01). "Bounding the diameter of distance-regular graphs". Combinatorica. 8 (4): 333–343. doi:10.1007/BF02189090. ISSN 0209-9683.