Equiangular lines via nodal domains

TL;DR

This paper investigates the maximum multiplicity of certain eigenvalues in bounded degree graphs, using nodal domain theory, and applies findings to improve bounds on equiangular lines.

Contribution

It introduces a novel approach using nodal domains to bound eigenvalue multiplicities, leading to improved results on equiangular lines.

Findings

Bound on eigenvalue multiplicity depending on maximum degree and cyclomatic number

Answer to Jiang et al.'s question for specific eigenvalues

Enhanced bounds on the number of equiangular lines

Abstract

For given $\Delta>0$ and $0<\lambda<3/\sqrt{2}$, we show that the maximum multiplicity that $\lambda$ can appear as the second largest eigenvalue of a connected graph with maximum degree at most $\Delta$ is $O_{\Delta,\lambda}(1)$. This result answers a question due to Jiang, Tidor, Yao, Zhang and Zhao [Question 6.4, Ann. of Math. (2) 194 (2021), no. 3, 729-743] in the case of $0<\lambda<3/\sqrt{2}$, and consequently leads to improvements in their results on equiangular lines. Our proof is based on the concept of nodal domains of eigenfunctions. Indeed, we establish a multiplicity estimate in terms of maximum degree and cyclomatic number of the graph, via a novel construction of eigenfunctions with large number of nodal domains.