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 θθ with multiplicity equal to their valency are studied. Let ΓΓ be such a graph. We first show that θ=−1θ=−1 if and only if ΓΓ is antipodal. Then we assume that the graph ΓΓ is primitive. We show that it is formally self-dual (and hence QQ-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∈VΓx,y∈VΓ be two adjacent vertices, and z∈Γ2(x)∩Γ2(y)z∈Γ2(x)∩Γ2(y). Then the intersection number τ2≔|Γ(z)∩Γ3(x)∩Γ3(y)|τ2≔|Γ(z)∩Γ3(x)∩Γ3(y)| is independent of the choice of vertices xx, yy and zz. In the case of the coset graph of the doubly truncated binary Golay code, we have b2=τ2b2=τ2. We classify all the graphs with b2=τ2b2=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.