Tails of bipartite distance-regular graphs

Let Gamma denote a bipartite distance-regular graph with diameter D greater than or equal to 4 and valency k greater than or equal to 3. Let theta(0) > theta(1) > ... > theta(D) denote the eigenvalues of Gamma and let E-0, E-1, ..., E-D denote the associated primitive idempotents. Fix s (1 less than or equal to s less than or equal to D-1) and abbreviate E := E-s. We say E is a tail whenever the entrywise product E o E is a linear combination of E0, E and at most one other primitive idempotent of Gamma. Let q(ij)(h) (0 less than or equal to h, i, j less than or equal to D) denote the Krein parameters of Gamma and let Delta denote the undirected graph with vertices 0, 1,..., D where two vertices i, j are adjacent whenever i = j and q(ij)(s) not equal 0. We show E is a tail if and only if one of (i)-(iii) holds: (i) A is a path; (ii) A has two connected components, each of which is a path; (iii) D = 6 and Delta has two connected components, one of which is a path on four vertices and the other of which is a clique on three vertices. (C) 2002 Elsevier Science Ltd. All rights reserved.