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 2\le i\le 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.