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 $D\ge3$, and eigenvalues $k=θ_0>θ_1>\cdots>θ_D$. Put $b\coloneqq b_1/(θ_1+1)$, and let $1=u_0>u_1>\cdots>u_D$ be the standard sequence associated with $θ_1$. 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 $-θ_D\le\prod_{i=2}^D(1-u_i)/(u_1-u_i) <(b+1)^{D-1}$. Equality in the product bound holds if and only if $Γ$ is $Q$-polynomial with respect to the primitive idempotent corresponding to $θ_1$. For every integer $D\ge4$, we further establish the explicit bound $-θ_D\le b^{D-1}+b^{D-2} +\frac{27}{2}(D-2)b^{D-3}$ whenever $b\ge\max\{8,\sqrt{12(D-2)}\}$. Both leading coefficients are best possible for each fixed $D\ge4$. The proof uses a spectral product theorem of Micchelli and Willoughby applied to Schur multiplication in the Bose--Mesner algebra.