The Q-polynomial idempotents of a distance-regular graph

We obtain the following characterization of Q-polynomial distance-regular graphs. Let Γ denote a distance-regular graph with diameter d ⩾ 3 . Let E denote a minimal idempotent of Γ which is not the trivial idempotent E 0 . Let { θ i ∗ } i = 0 d denote the dual eigenvalue sequence for E. We show that E is Q-polynomial if and only if (i) the entry-wise product E ∘ E is a linear combination of E 0 , E, and at most one other minimal idempotent of Γ; (ii) there exists a complex scalar β such that θ i − 1 ∗ − β θ i ∗ + θ i + 1 ∗ is independent of i for 1 ⩽ i ⩽ d − 1 ; (iii) θ i ∗ ≠ θ 0 ∗ for 1 ⩽ i ⩽ d .