Distance-regular graph

In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two verticesv and w, the number of vertices at distancej from v and at distance k from w depends only upon j, k, and the distance between v and w.

Some authors exclude the complete graphs and disconnected graphs from this definition.

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.