报告名称：Multiplicity of the second-largest eigenvalue of graphs

报告专家：Guantao Chen

专家所在单位：美国佐治亚州立大学

报告时间：2023年12月28日

报告地点: 英国威廉希尔体育公司203会议室

专家简介：陈冠涛，佐治亚州立大学教授，数学与统计系主任。主要研究方向为图论及其应用。主要研究图的结构问题，如图的圈和路、图染色和图的Ramsey理论。近年来，他的主要工作是研究对图的边进行重新着色的技巧，并利用它们解决该领域的一些经典问题。他在组合学和图论的主要期刊上发表了120多篇论文，并与他的许多合作者解决了许多长期存在的猜想。曾担任SIAM离散数学活动组的组织者(2014-2016)，以及《图与组合学》杂志的执行主编(2011年以来)。

报告摘要：The multiplicity of the second-largest eigenvalue of the adjacency matrix A(G) of a connected graph G, denoted by m(λ 2 ,G), is the number of times of the second-largest eigenvalue of A(G) appears. In 2019, Jiang, Tidor, Yao, Zhang and Zhao gave an upper bound on m(λ 2 ,G) for graphs G with bounded degrees, and applied it to solve a longstanding problem on equiangular lines. We showed that if G is a 3-connected planar graph or 2-connected outerplanar graph, then m(λ 2 ,G) ≤ δ(G), where δ(G) is the minimum degree of G. We further prove that if G is a connected planar graph, then m(λ 2 ,G) ≤ ∆(G); if G is a connected outerplanar graph, then m(λ 2 ,G) ≤ max{2,∆(G) − 1}, where ∆(G) is the maximum degree of G. Moreover, these two upper bounds for connected planar graphs and outerplanar graphs, respectively, are best possible. We will discuss general techniques and specific methods we used in the proofs of these results.