Resolving a Problem of Lv and Koolen on the Smallest Eigenvalue of Distance-Regular Graphs
Abstract
Let Γ be a distance-regular graph with valency k, diameter D≥3, and eigenvalues k=θ0>θ1>⋯>θD. Put bcoloneqq 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.
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BibTeX
@misc{hong2026resolving,
title = {{Resolving a Problem of Lv and Koolen on the Smallest Eigenvalue of Distance-Regular Graphs}},
author = {Zhen-Mu Hong and Fu-Tao Hu and Chenhui Lv},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37010},
eprint = {2610.37010},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37010},
}
RIS
TY - UNPB
TI - Resolving a Problem of Lv and Koolen on the Smallest Eigenvalue of Distance-Regular Graphs
AU - Zhen-Mu Hong
AU - Fu-Tao Hu
AU - Chenhui Lv
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37010
UR - https://arxiv.org/abs/2610.37010
AB - Let Γ be a distance-regular graph with valency k, diameter D≥3, and eigenvalues k=θ_0>θ_1>⋯>θ_D. Put bcoloneqq b_1/(θ_1+1), and let 1=u_0>u_1>⋯>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≤∏_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≥4, we further establish the explicit bound −θ_D≤ b^D−1+b^D−2 +27/2(D−2)b^D−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.
ER -