A characterization of Q-polynomial distance-regular graphs

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 E0. Let {θ∗ i }i=0 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 E0, E, and at most one other minimal idempotent of Γ; (ii) there exists a complex scalar β such that θ∗ i−1 − βθ∗ i + θ∗ i+1 is independent of i for 1 ≤ i ≤ d− 1; (iii) θ∗ i 6= θ∗ 0 for 1 ≤ i ≤ d.