Almost 2-homogeneous Bipartite Distance-regular Graphs

Let Γ = (X, R) denote a bipartite distance-regular graph with diameter d ≥ 4, and fix a vertex x of Γ. The Terwilliger algebra of Γ with respect to x is the subalgebra T ofMatX (C) generated by A, E * 0,E * 1,⋯ , E * d, where A is the adjacency matrix ofΓ , and where E * i denotes the projection onto the i th subconstituent ofΓ with respect to x. Let W denote an irreducible T -module. W is said to be thin whenever dim E * iW ≤ 1 (0 ≤ i ≤ d). The endpoint of W is min{ i |E * iW ≠ = 0}. It is known that a thin irreducibleT -module of endpoint 2 has dimension d − 3, d − 2, ord − 1. Γ is said to be 2-homogeneous whenever for all i(1 ≤ i ≤ d − 1 ) and for all x, y,z ∈ X with ∂(x, y) = 2,∂ (x, z) = i, ∂(y,z ) = i, the number | Γ1(x) ∩ Γ1(y) ∩ Γi − 1(z)| is independent ofx , y, z. Nomura has classified the 2-homogeneous bipartite distance-regular graphs. In this paper we study a slightly weaker condition. Γ is said to be almost 2-homogeneous whenever for all i(1 ≤ i ≤ d − 2 ) and for all x, y,z ∈ X with ∂(x, y) = 2,∂ (x, z) = i, ∂(y,z ) = i, the number | Γ1(x) ∩ Γ1(y) ∩ Γi − 1(z)| is independent ofx , y, z. We prove that the following are equivalent: (i)Γ is almost 2-homogeneous; (ii)Γ has, up to isomorphism, a unique irreducible T -module of endpoint 2 and this module is thin. Moreover, Γ is 2-homogeneous if and only if (i) and (ii) hold and the unique irreducible T -module of endpoint 2 has dimension d − 3.