Almost 2-homogeneous Bipartite Distance-regular Graphs

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 of MatX (C) generated by A, E*0, E*1, . . . , E*d, where A is the adjacency matrix of Γ, and where E*i denotes the projection onto the ith subconstituent of Γ with respect to x. Let W denote an irreducible T-module. W is said to be thin whenever dim E*i W ≤ 1 (0 ≤ i ≤ d). The endpoint of W is min{i | E*i W ≠ 0}. It is known that a thin irreducible T-module of endpoint 2 has dimension d - 3, d - 2, or d - 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 |ΓF1(x) ∩ Γ1 (y) ∩ Γi-1 (z)| is independent of x, 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 of x, 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. © 2000 Academic Press.