讲座内容：

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.

References:

[1]Lele Liu, Bo Ning, Local properties of the spectral radius and

Perron vector in graphs, J. Combin. Theory Ser. B 176 (2026), 241–253.

[2]Binlong Li, Bo Ning, Localized and weighted versions of

extremal problems, arXiv:2509.17055.

[3]Lele Liu, Bo Ning, A new spectral Tur\'an theorem for weighted

graphs and consequences, arXiv:2510.26410.