On the Terwilliger algebra of bipartite distance-regular graphs with Δ2=0 and c2=1

Let Gamma denote a bipartite distance-regular graph with diameter D >= 4 and valency k >= 3. Let X denote the vertex set of Gamma, and let A denote the adjacency matrix of Gamma. For x is an element of X and for 0 <= i <= D, let Gamma(i)(x) denote the set of vertices in X that are distance i from vertex x. Define a parameter Delta(2) in terms of the intersection numbers by Delta(2) = (k-2) (c(3)-1) - (c(2)-1)p(22)(2) . We first show that Delta(2) = 0 implies that D <= 5 or c(2) is an element of {1, 2}. For x is an element of X let T = T(x) denote the subalgebra of Mat(X)(C) generated by A, E-0*, E-0*,..., E-D*, where for 0 <= i <= D, E-i* represents the projection onto the ith subconstituent of Gamma with respect to x. We refer to T as the Terwilliger algebra of Gamma with respect to x. By the endpoint of an irreducible T-module W we mean min {i vertical bar E-i*W not equal 0}. In this paper we assume Gamma has the property that for 2 <= i <= D - 1, there exist complex scalars alpha(i), beta(i) such that for all x, y, z is an element of X with partial derivative(x, y) = 2, partial derivative(x, z) = i, partial derivative(y, z) = i, we have alpha(i) + beta(i)vertical bar Gamma(1)(x) boolean AND Gamma(1)(y) boolean AND Gamma(i-1)(z)vertical bar = vertical bar Gamma(i-1)(x) boolean AND Gamma(i-1)(y) boolean AND Gamma(1)(z)vertical bar. We additionally assume that Delta(2) = 0 with c(2) = 1. Under the above assumptions we study the algebra T. We show that if Gamma is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2. We give an orthogonal basis for this T-module, and we give the action of A on this basis. (C) 2016 Elsevier Inc. All rights reserved.