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

In this paper, triangle-free distance-regular graphs with diameter 3 and an eigenvalue θ with multiplicity equal to their valency are studied. Let Γ be such a graph. We first show that θ = −1 if and only if Γ is antipodal. Then we assume that the graph Γ is primitive. We show that it is formally self-dual (and hence Q-polynomial and 1-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 ∈ VΓ be two adjacent vertices, and z ∈ Γ2(x) ∩ Γ2(y). Then the intersection number τ2 := |Γ (z) ∩ Γ3(x) ∩ Γ3(y)| 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 b2 = τ2. We classify all the graphs with b2 = τ2 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.