Equiangular lines via nodal domains

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