Triangle-free distance-regular graphs with an eigenvalue multiplicity equal to their valency and diameter 3

Abstract

In this paper, triangle-free distance-regular graphs with diameter 3 and an eigenvalue theta with multiplicity equal to their valency are studied. Let Gamma be such a graph. We first show that theta = -1 if and only if Gamma is antipodal. Then we assume that the graph Gamma is primitive. We show that it is formally self-dual (and hence Q-polynomial and I-homogeneous), all its eigenvalues are integral, and the eigenvalue with multiplicity equal to the valency is either second largest or the smallest. Let x, y is an element of V Gamma be two adjacent vertices, and z is an element of Gamma(2)(x) boolean AND Gamma(2)(y). Then the intersection number tau(2) := vertical bar Gamma(z) boolean AND Gamma(3)(x) boolean AND Gamma(3)(y)vertical bar is independent of the choice of vertices x, y and z. In the case of the coset graph of the doubly truncated binary Golay code, we have b(2) = tau(2). We classify all the graphs with b(2) = tau(2) and establish that the just mentioned graph is the only example. In particular, we rule out an infinite family of otherwise feasible intersection arrays. (c) 2006 Elsevier Ltd. All rights reserved.