On bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint two

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 P. For x is an element of 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 ith subconstituent of Gamma with respect to x. We refer to T as the Terwilliger algebra of Gamma with respect to x. An irreducible T-module W is said to be thin whenever dim E-i* <= 1 for 0 <= i <= D. By the endpoint of W we mean min{|E-i*W not equal 0}. For 0 <= i <= D, let Gamma(i)(z) denote the set of vertices in X that are distance i from vertex z. 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). In this paper we prove the following are equivalent: (i) Delta(2) > 0 and for 2 <= i < D - 2 there exist complex scalars alpha(i) beta(i) with the following property: for all x, y, z is an element of X such that partial derivative(x, y) = 2, partial derivative(x, z) = i, partial derivative(y, z) = i we have alpha(i) + beta(i)|Gamma(1)(x)boolean AND Gamma(1)(Y)boolean AND Gamma(i-1)(Z)| = |Gamma(i-1)(x)boolean AND Gamma(i-1)(y)boolean AND Gamma(1)(z)| (ii) For all x is an element of X there exist up to isomorphism exactly two irreducible modules for the Terwilliger algebra T(x) with endpoint two, and these modules are thin. (C) 2016 Elsevier Inc. All rights reserved.