Tight Distance-Regular Graphs and the Q-Polynomial Property

Abstract. 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.