Resolving a Problem of Lv and Koolen on the Smallest Eigenvalue of Distance-Regular Graphs

Let $Γ$ be a distance-regular graph with valency k, diameter D≥3, and eigenvalues k=θ₀>θ₁>⋯>θD. Put b b₁/(θ₁+1), and let 1=u₀>u₁>⋯>uD be the standard sequence associated with θ₁. Lv and Koolen asked whether the negative of the smallest eigenvalue of a distance-regular graph can be bounded above in terms of its diameter D and the parameter b alone. In this paper, we answer this question affirmatively by proving that -θD≤∏ᵢ₌₂D(1-uᵢ)/(u₁-uᵢ) <(b+1)D-1. Equality in the product bound holds if and only if $Γ$ is Q-polynomial with respect to the primitive idempotent corresponding to θ₁. For every integer D≥4, we further establish the explicit bound -θD≤ bD-1+bD-2 +27/2(D-2)bD-3 whenever b≥max\8,√12(D-2)\. Both leading coefficients are best possible for each fixed D≥4. The proof uses a spectral product theorem of Micchelli and Willoughby applied to Schur multiplication in the Bose--Mesner algebra.