This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.

Distance-regular graphs

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The vertex-connectivity of a distance-regular graph

We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs.

Eigenvalues of Cayley graphs

Distance-regular graphs with a1 or c2 at least half the valency

A resolving set for a graph G is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of G is the smallest size of a resolving set for G. A graph is distance-regular if, for any two vertices u; v at each distance i, the number of neighbours of v at each possible distance from u (i.e. i 1, i or i+ 1) depends only on the distance i, and not on the choice of vertices u; v. The class of distance-regular graphs includes all distance-transitive graphs and all strongly regular graphs. In this paper, we present the results of computer calculations which have found the metric dimension of all distance-regular graphs on up to 34 vertices, low-valency distance transitive graphs on up to 100 vertices, strongly regular graphs on up to 45 vertices, rank-3 strongly regular graphs on under 100 vertices, as well as certain other distance-regular graphs.

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The metric dimension of small distance-regular and strongly regular graphs

We show that each distance-regular graph of valency four has known parameters.

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J. H. Koolen

Papers

The Distance-Regular Graphs of Valency Four