Abstract

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)  ∩ Γ_i − 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)  ∩ Γ_i − 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.