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,yVΓ be two adjacent vertices, and zΓ2(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.
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