报告标题:An $A_\alpha$-spectral Erd\H{o}s-S\'os theorem and some related problems
报告人：李书超教授 华中师范大学
报告时间：2022/11/16（周三），9:00-9:40
报告地点：腾讯会议455 861 864
报告摘要:
Let $G$ be a connected graph and let $\alpha$ be a real number in $[0,1].$ In 2017, Nikiforov proposed the $A_\alpha$-matrix for $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. Let $S_{n,k}$ be the complete splitting graph, which is the join of the graphs $K_k$ and $n-k$ isolated vertices. The famous Erd\H{o}s-S\'os conjecture states that every $n$-vertex graph with more than $\frac{1}{2}(k-1)n$ edges must contain every tree on $k+1$ vertices; The Nikiforov's conjecture states that, for fixed $k\geq2,$ and sufficiently large $n$, the $\{C_{2k+1},C_{2k+2}\}$-free graph of order $n$ with maximum adjacency spectral radius is $S_{n,k}$ and the $C_{2k+2}$-free graph of order $n$ with maximum adjacency spectral radius is $S^+_{n,k}.$ The famous Erd\H{o}s-P\'osa theorem shows that, for $k\geq2$ and $n\geq24k,$ every $n$-vertex graph $G$ with at least $(2k-1)(n-k)$ edges contains $k$ independent cycles, unless $G\cong S_{n,2k-1}.$ In this talk, by a unified approach we give an $A_\alpha$-spectral version for the Erd\H{o}s-S\'os conjecture, Nikiforov conjecture, and Erd\H{o}s-P\'osa theorem, respectively. This is a joint work with M.Z. Chen, Z.M. Li, Y.T. Yu, H.H. Zhang and X.D. Zhang.
报告人简介：李书超，理学博士，华中师范大学教授，博士生导师，主要从事组合数学、图论及其应用方面的研究。相继在Advances in Applied Mathematics, Journal of Algebraic Combinatorics, European Journal of Combinatorics，Discrete Mathematics等学术期刊发表论文100余篇。先后主持、完成国家自科项目4项，其中在研项目一项；主持科技部项目3项。2013年皇冠hg8868老版本入选教育部“新世纪优秀人才支持计划”。主持完成的课题“图的几类不变量的研究”获湖北省自然科学奖。
