Tight distance-regular graphs and the Q-polynomial property

Let Γ denote a distance-regular graph with diameter d ≥ 3, and assume Γ is tight (in the sense of Jurišić, Koolen and Terwilliger). Let θ denote the second largest or smallest eigenvalue of Γ, and let σ0, σ1, . . . , σd denote the associated cosine sequence. We obtain an inequality involving σ0, σ1, . . . , σd for each integer i (1 ≤ i ≤ d - 1), and we show equality for all i is closely related to Γ being Q-polynomial with respect to θ. We use this idea to investigate the Q-polynomial structures in tight distance-regular graphs. © Springer-Verlag 2001