Triangle-free distance-regular graphs

Let Γ = (X, R) denote a distance-regular graph with distance function ∂ and diameter d ≥ 3. For 2 ≤ i ≤ d, by a parallelogram of length i, we mean a 4-tuple xyzu of vertices in X such that ∂(x, y) = ∂(z, u) = 1, ∂(x, u) = i, and ∂(x, z) = ∂(y, z) = ∂(y, u) = i − 1. Suppose the intersection number a1 = 0, a2 6= 0 in Γ. We prove the following (i)-(ii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of length 3; (ii) Γ has classical parameters. By applying the above result we show that if Γ has classical parameters and the intersection numbers a1 = 0, a2 6= 0, then for each pair of vertices v, w ∈ X at distance ∂(v, w) = 2, there exists a strongly regular subgraph Ω of Γ containing v, w. Furthermore, for each vertex x ∈ Ω, the subgraph induced on Ω2(x) is an a2-regular connected graph with diameter at most 3.