A new inequality for distance-regular graphs Original Research Article

Given a nontrivial primitive idempotent E of a distance-regular graph Γ with diameter d ⩾ 3, we obtain an inequality involving the intersection numbers of Γ for each integer i (3 ⩽ i ⩽ d). We show equality is attained for i = 3 if and only if equality is attained for all i (3 ⩽ i ⩽ d) if and only if Γ is Q-polynomial with respect to E. If the intersection numbers of Γ are such that qci − bi − q(qci−1 − bi−1) is independent of i (1 ⩽ i ⩽ d) for some q∈∝\{0, −1} (as is the case for many examples), our inequalities take an especially simple form.