On bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint two.

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 let T = T ( x ) denote the subalgebra of Mat X ( C ) generated by A , E 0 ⁎ , E 1 ⁎ , … , E D ⁎ , where for 0 ≤ i ≤ D , E i ⁎ represents the projection onto the i th subconstituent of Γ with respect to x . We refer to T as the Terwilliger algebra of Γ with respect to x . An irreducible T -module W is said to be thin whenever dim E i ⁎ W ≤ 1 for 0 ≤ i ≤ D . By the endpoint of W we mean min { i | E i ⁎ W ≠ 0 } . For 0 ≤ i ≤ D , let Γ i ( z ) denote the set of vertices in X that are distance i from vertex z . Define a parameter Δ 2 in terms of the intersection numbers by Δ 2 = ( k − 2 ) ( c 3 − 1 ) − ( c 2 − 1 ) p 22 2 . In this paper we prove the following are equivalent: (i) Δ 2 > 0 and for 2 ≤ i ≤ D − 2 there exist complex scalars α i , β i with the following property: for all x , y , z ∈ X such that ∂ ( x , y ) = 2 , ∂ ( x , z ) = i , ∂ ( y , z ) = i we have α i + β i | Γ 1 ( x ) ∩ Γ 1 ( y ) ∩ Γ i − 1 ( z ) | = | Γ i − 1 ( x ) ∩ Γ i − 1 ( y ) ∩ Γ 1 ( z ) | ; (ii) For all x ∈ X there exist up to isomorphism exactly two irreducible modules for the Terwilliger algebra T ( x ) with endpoint two, and these modules are thin. [ABSTRACT FROM AUTHOR]