On bipartite distance-regular graphs with exactly one non-thin [formula omitted]-module 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 and for 0 ≤ i ≤ D , let Γ i ( x ) denote the set of vertices in X that are distance i from vertex x . 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 . 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 an irreducible T -module W we mean min { i | E i ∗ W ≠ 0 } . Fix x ∈ X and assume that Γ has, up to isomorphism, exactly one irreducible T -module W with endpoint 2 , and that W is non-thin with dim ( E 2 ∗ W ) = 1 , dim ( E D − 1 ∗ W ) ≤ 1 and dim ( E i ∗ W ) ≤ 2 for 3 ≤ i ≤ D . We prove that for 2 ≤ i ≤ D , there exist complex scalars α i , β i such that | Γ i − 1 ( x ) ∩ Γ i − 1 ( y ) ∩ Γ 1 ( z ) | = α i + β i | Γ 1 ( x ) ∩ Γ 1 ( y ) ∩ Γ i − 1 ( z ) | for all y ∈ Γ 2 ( x ) and z ∈ Γ i ( x ) ∩ Γ i ( y ) . Furthermore, we prove Δ 2 = 0 and either D = 5 or c 2 ∈ { 1 , 2 } . We show there exist integers 3 ≤ f ≤ ℓ ≤ D − 2 such that dim ( E i ∗ W ) = 2 if and only if f ≤ i ≤ ℓ . [ABSTRACT FROM AUTHOR]