Abstract
Let Γ denote a distance-regular graph with diameter d⩾3. Let E, F denote nontrivial primitive idempotents of Γ such that F corresponds to the second largest or the least eigenvalue. We investigate the situation that there exists a primitive idempotent H of Γ such that E∘ 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 Γ being Q-polynomial with respect to E. This generalizes a result of Lang on bipartite graphs and a result of Pascasio on tight graphs.
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