On the primitive idempotents of distance-regular graphs

Let Gamma denote a distance-regular graph with diameter d greater than or equal to 3. Let E, F denote nontrivial primitive idempotents of Gamma such that F corresponds to the second largest or the least eigenvalue. We investigate the situation that there exists a primitive idempotent H of Gamma such that E circle F is a linear combination of F and H. Our main purpose is to obtain the inequalities involving the cosines of E, and to show that equality is closely related to Gamma being Q-polynomial with respect to E. This generalizes a result of Lang on bipartite graphs and a result of Pascasio on tight graphs. (C) 2001 Elsevier Science B.V. All rights reserved.