A valency bound for distance-regular graphs

Abstract

The regular complete t-partite graphs Kt×s (s,t positive integers at least 2) with valency k=(t−1)s have smallest eigenvalue −s=−k/(t−1), and hence, for fixed t there are infinitely many of them. In this paper we will show that these graphs are exceptional graphs for the class of distance-regular graphs. For this we will show a valency bound for distance-regular graphs with a relatively large, in absolute value, smallest eigenvalue. Using this bound, we classify the non-bipartite distance-regular graphs with diameter at most three with smallest eigenvalue not larger than −k/2, where k is the valency of the graph. As an application we complete the classification of the 3-chromatic distance-regular graphs with diameter three, which was started by Blokhuis, Brouwer and Haemers.