On the Terwilliger algebra of bipartite distance-regular graphs with [formula omitted] and [formula omitted

Abstract

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 and for 0≤i≤D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. It is known that Δ2=0 implies that D≤5 or c2∈{1,2}.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. By the endpoint of an irreducible T-module W we mean min{i|Ei∗W≠0}.We find the structure of irreducible T-modules of endpoint 2 for graphs Γ which have the property that for 2≤i≤D−1, there exist complex scalars αi, βi such that for all x,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)|, in case when Δ2=0 and c2=2. The case when Δ2=0 and c2=1 is already studied by MacLean et al. [15].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, and we give the action of A on this basis.