On the Terwilliger algebra of bipartite distance-regular graphs with Delta_2=0 and c_2=2

Let Gamma denote a bipartite distance-regular graph with diameter D ge 4 and valency k ge 4. Let X denote the vertex set of Gamma, and let A denote the adjacency matrix of Gamma. For x in X and for 0 le i le D, let G_i(x) denote the set of vertices in X that are distance i from vertex x. Define a parameter Delta_2 in terms of the intersection numbers by Delta_2 = (k-2)(c_3-1)-(c_2-1)p^2_{22}. It is known that Delta_2 = 0 implies that D le 5 or c_2 in {1,2}. 

For x in X let T=T(x) denote the subalgebra of Mat_X(C) generated by A, Es_0, Es_1, …, Es_D, where for 0 le i le D, Es_i represents the projection onto the i-th subconstituent of Gamma with respect to x. We refer to T as the Terwilliger algebra of Gamma with respect to x. By the endpoint of an irreducible T-module W we mean min{i | Es_iW ne 0}.  

Building on the work of [MacLean, Miklavič and Penjić: On the Terwilliger algebra of bipartite distance-regular graphs with Delta_2=0 and c_2=1, Linear Algebra and its Applications 496 (2016)] we find the structure of nonthin irreducible T-modules of endpoint 2 for graphs Gamma in which for 2le ile D-1 there exist complex scalars alpha_i, beta_i with the following property: for all x, y, z in X such that partial(x, y) = 2, partial(x, z) = i,  partial(y, z) = i we have alpha_i + beta_i  |G_1(x) cap G_1(y) cap G_{i-1}(z)| = |G_{i-1}(x) cap G_{i-1}(y) cap G_1(z)|; in case when Delta_2=0 and c_2=2.

We show that if Gamma 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 an basis for this T-module, and we give the action of A on this basis. The isomorphism class of such a given module is determined by the intersection numbers of the Gamma.