On bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint two

Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and let A denote the adjacency matrix of Γ. For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A, E0⁎,E1⁎,…,ED⁎, where for 0≤i≤D, Ei⁎ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim Ei⁎W≤1 for 0≤i≤D. By the endpoint of W we mean min{i|Ei⁎W≠0}. For 0≤i≤D, let Γi(z) denote the set of vertices in X that are distance i from vertex z. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. In this paper we prove the following are equivalent: (i) Δ2>0 and for 2≤i≤D−2 there exist complex scalars αi,βi with the following property: for all x,y,z∈X such that ∂(x,y)=2, ∂(x,z)=i, ∂(y,z)=i we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|; (ii) For all x∈X there exist up to isomorphism exactly two irreducible modules for the Terwilliger algebra T(x) with endpoint two, and these modules are thin.