We obtain the following characterization of Q -polynomial distanceregular graphs. Let Γ denote a distance-regular graph with diam-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 − βθ * i + θ * i+1 is independent of i for 1 i d − 1; (iii) θ * i = θ * 0 for 1 i d.
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