标题： 等角线通过节点域 显示英文标题

标题： Equiangular lines via nodal domains

摘要： 对于给定的$\Delta>0$和$0<\lambda<3/\sqrt{2}$，我们证明了$\lambda$作为最大度数不超过$\Delta$的连通图的第二大特征值的最大重数是$O_{\Delta,\lambda}(1)$。这个结果回答了 Jiang、Tidor、Yao、Zhang 和 Zhao [《数学年刊》(2) 194 (2021), 第3期, 729-743页, 问题6.4] 在$0<\lambda<3/\sqrt{2}$情况下的问题，并由此导致了他们在等角线方面的结果的改进。我们的证明基于特征函数的节点域的概念。事实上，我们通过构造具有大量节点域的特征函数，建立了一个关于图的最大度数和环量数的重数估计。 显示英文摘要

摘要： 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.