Q-polynomial distance-regular graphs with a(1)=0 and a(2) not equal 0

Let Gamma denote a Q-polynomial distance-regular graph with diameter D >= 3 and intersection numbers a(1) = 0, a(2) not equal 0. Let X denote the vertex set of Gamma and let A is an element of Mat(X) (C) denote the adjacency matrix of Gamma. Fix x is an element of X and let A* E Mat(X)(C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra, of Mat(X) (C) generated by A, A*. We call T the Terwilliger algebra of Gamma with respect to x. We show that up to isomorphism there exists a unique irreducible T-module W with endpoint 1. We show that W has dimension 2D - 2. We display a basis for W which consists of eigenvectors for A*. We display the action of A on this basis. We show that W appears in the standard module of Gamma with multiplicity k - 1, where k is the valency of Gamma. (C) 2008 Published by Elsevier Ltd