The Q -polynomial idempotents of a distance-regular graph

We obtain the following characterization of Q -polynomial distance-regular graphs. Let Gamma denote a distance-regular graph with diameter d >= 3. Let Gamma denote a minimal idempotent of Gamma which is not the trivial idempotent E(0). Let (0(i)*)(i)(d)=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 o E is a linear combination of E(0). E. and at most one other minimal idempotent of Gamma; (ii) there exists a complex scalar beta such that 0*(i-1) beta 0*(i) + 0*(i+1), is independent of i for 1 <= i <= d - 1: (iii) 0(i)* not equal 0(0)* for 1 <= i <= d. (C) 2010 Elsevier Inc. All rights reserved.