Extremal spectral results of planar graphs without vertex-disjoint cycles

报告学者:林辉球 教授




报告摘要: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.

报告者简介:林辉球,华东理工大学教授,博士生导师。从事图论研究,特别是图谱理论、极值图论研究。先后主持国家自然科学基金项目3项; 在J. Combin. Theory Ser. BJ Graph Theory、European J Combin,Combin. Probab. Comput等期刊上发表学术论文90余篇。