On the Terwilliger algebra of certain family of bipartite distance-regular graphs with Δ_2 = 0

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 Ai (0 ≤ i ≤ D) denote the distance matrices of Γ. We abbreviate A := A1. For x ∈ X and for 0 ≤ i ≤ D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. Fix x ∈ X and 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. By the endpoint of an irreducible T-module W we mean min{i ∣ Ei*W ≠ 0}. In this paper we assume Γ has the property that for 2 ≤ i ≤ D − 1, there exist complex scalars αi, βi such that for all y, z ∈ X with ∂(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)|. We study the structure of irreducible T-modules of endpoint 2. Let W denote an irreducible T-module with endpoint 2, and let v denote a nonzero vector in E2*W. We show that W = span({Ei*Ai − 2E2*v ∣ 2 ≤ i ≤ D} ∪ {Ei*Ai + 2E2*v ∣ 2 ≤ i ≤ D − 2}). It turns out that, except for a particular family of bipartite distance-regular graphs with D = 5, this result is already known in the literature. Assume now that Γ is a member of this particular family of graphs. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2 and it is not thin. We give a basis for this T-module.