报告题目：Localized and weighted versions of extremal problems

报 告 人：宁博 教授 南开大学

报告时间：2026年6月26日下午2：30

报告地点：数学研究中心报告厅

报告摘要 In this talk, we discuss various extensions of Tur\'an-type problems, including localized, weighted, and their connections. First, we provide short and new proofs for recent theorems by Malec--Tompkins~(EUJC, 2023) and Zhao--Zhang~(JGT, 2025). We also give a short proof of a weighted localized Tur\'an-type result for paths, which relies on the \emph{Small Path Double Cover Conjecture}. Furthermore, we present localized versions of the Balister--Bollob\'as--Riordan--Schelp theorem on paths and the Erd\H{o}s--Gallai theorem on matchings. Our second focus is on local versions of spectral Tur\'an theorems. We first show that for any graph $G$, $\lambda(G)^2 \le 2 \sum_{e \in E(G)} \frac{\cl(e)-1}{\cl(e)},$ where $\cl(e)$ denotes the order of the largest clique containing edge $e$. We then extend this inequality to weighted graphs: $\lambda(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\cl(e) - 1}{\cl(e)} w(e)^2$.

This main theorem leads to several new results, including two vertex-based andvertex-degree-based local Tur\'an theorems, weighted generalizations of the Edwards--Elphick and Cvetkovi\'c theorems, and two localized versions of Wilf’s theorem. Notably, one of the localized Wilf-type results confirms a conjecture from probability and operator algebras, independently posed by R.~Tripathi. Additionally, our theorem unifies and implies many earlier results from spectral andextremal graph theory.

报告人简介：宁博，南开大学计算机学院、密码与网络空间安全学院教授，博士生导师。国家高层次青年人才（教育部青年长江学者），北洋青年学者，南开大学百青（A类），南洋理工大学、日本横滨国立大学、匈牙利Renyi数学所等访问学者。在图论领域顶级期刊Combinatorica/JCTB , 权威期刊JGT,SIDMA,CPC和综合性期刊Canadian J Math，Adv. Appl. Math.等发表论文 60 余篇。主持国家级项目4项，参与科技部重点研发计划2项。现主持面上项目和天元访学项目（导师）各1项。代表性工作是合作解决了经典图论教科书《Graph Theory with Applications》附录中列出的图论领域 50个未解决问题的问题7。曾受邀在第九届世界华人数学家大会作45分钟特邀报告，并获第八届中国运筹学会青年科技奖、陕西省高校科学技术研究优秀成果一等奖。