On the Terwilliger algebra of bipartite distance-regular graphs with Δ2 = 0 and c2 = 1

Let Γ denote a bipartite distance-regular graph with diameter D≥4D≥4 and valency k≥3k≥3. Let X denote the vertex set of Γ, and let A   denote the adjacency matrix of Γ. For x∈Xx∈X and for 0≤i≤D0≤i≤D, let Γi(x)Γi(x) denote the set of vertices in X that are distance i from vertex x  . Define a parameter Δ2Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. We first show that Δ2=0Δ2=0 implies that D≤5D≤5 or c2∈{1,2}c2∈{1,2}.For x∈Xx∈X let T=T(x)T=T(x) denote the subalgebra of MatX(C)MatX(C) generated by A,E0⁎,E1⁎,…,ED⁎, where for 0≤i≤D0≤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−12≤i≤D−1, there exist complex scalars αiαi, βiβi such that for all x,y,z∈Xx,y,z∈X with ∂(x,y)=2∂(x,y)=2, ∂(x,z)=i∂(x,z)=i, ∂(y,z)=i∂(y,z)=i, we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|. We additionally assume that Δ2=0Δ2=0 with c2=1c2=1.Under the above assumptions we study the algebra T. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2. We give an orthogonal basis for this T-module, and we give the action of A on this basis.