Equiangular lines via nodal domains

Abstract

For given $\hat{I}”>0$ and $0<\hat{I}»<3/\sqrt{2}$, we show that the maximum multiplicity that $\hat{I}»$ can appear as the second largest eigenvalue of a connected graph with maximum degree at most $\hat{I}”$ is $O_{\hat{I}”,\hat{I}»}(1)$. 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 $0<\hat{I}»<3/\sqrt{2}$, 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.

10 pages, 1 figure