Distance-regular graphs with a(1) or c(2) at least half the valency

Abstract

In this paper, we study the distance-regular graphs Gamma that have a pair of distinct vertices, say x and y. such that the number of common neighbors of x and y is about half the valency of Gamma. We show that if the diameter is at least three, then such a graph, besides a finite number of exceptions, is a Taylor graph, bipartite with diameter three or a line graph. (C) 2011 Elsevier Inc. All rights reserved.