报告地点：行健楼学术活动室665
Abstract：Given a planar graph family $\mathcal{F}$, let ${\rm ex}_{\mathcal{P}}(n,\mathcal{F})$ and ${\rm spex}_{\mathcal{P}}(n,\mathcal{F})$ be the maximum size and maximum spectral radius over all $n$-vertex $\mathcal{F}$-free planar graphs, respectively.
Let $tC_k$ be the disjoint union of $t$ copies of $k$-cycles, and $t\mathcal{C}$ be the family of $t$ vertex-disjoint cycles without length restriction. Tait and Tobin [Three conjectures in extremal spectral graph theory, J. Combin. Theory Ser. B 126 (2017) 137--161] determined that $K_2+P_{n-2}$ is the extremal spectral graph among all planar graphs with sufficiently large order $n$, which implies the extreme graphs of $spex_{\mathcal{P}}(n,tC_{\ell})$ and $spex_{\mathcal{P}}(n,t\mathcal{C})$ for $t\geq 3$ are $K_2+P_{n-2}$. In this paper, we first determine $spex_{\mathcal{P}}(n,tC_{\ell})$ and $spex_{\mathcal{P}}(n,t\mathcal{C})$ and characterize the unique extremal graph for $1\leq t\leq 2$, $\ell\geq 3$ and sufficiently large $n$.
Secondly, we obtain the exact values of ${\rm ex}_{\mathcal{P}}(n,2C_4)$ and ${\rm ex}_{\mathcal{P}}(n,2\mathcal{C})$, which answers a conjecture of Li [Planar Tur\'an number of disjoint union of $C_3$ and $C_4$, Discrete Appl. Math. 342 (2024) 260-274]. These present a new exploration of approaches and tools to investigate extremal problems of planar graphs.This is a joint work with Longfei Fang and Yongtang Shi.
报告人简介：林辉球，华东理工大学数学学院副院长（主持工作）、教授、博士生导师，上海市东方学者特聘教授，2013年博士毕业于华东师范大学。中国运筹学会图论组合分会理事。在图论的主流期刊《J. Combin. Theory, Series B》、《Combin. Probab. Comput.》、《J. Graph Theory》、《European J. Comb.》、《Eletrionic J. Comb.》等发表学术论文60余篇。主持国家自然科学基金项目4项，目前主持在研国家自然科学基金面上项目。
