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 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 ∈ V Γ be two adjacent vertices, and z ∈ Γ 2 ( x ) ∩ Γ 2 ( y ) . Then the intersection number τ 2 ≔ | Γ ( z ) ∩ Γ 3 ( x ) ∩ Γ 3 ( 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 b 2 = τ 2 . We classify all the graphs with b 2 = τ 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.