Bounding the diameter of distance-regular graphs

Abstract

Let G be a connected distance-regular graph with valency k>2 and diameter d, but not a complete multipartite graph. Suppose that θ is an eigenvalue of G with multiplicity m and that θ≠±k. We prove that both d and k are bounded by functions of m. This implies that, if m>1 is given, there are only finitely many connected, co-connected distance-regular graphs with an eigenvalue of multiplicity m. © 1988 Akadémiai Kiadó.