A new spectral Tur\'an theorem for weighted graphs and consequences

TL;DR

This paper extends spectral Turán theorems to weighted graphs, providing new bounds, characterizations, and numerous implications for extremal and spectral graph theory, unifying and strengthening previous results.

Contribution

It introduces a weighted graph inequality generalizing earlier spectral bounds, characterizes extremal graphs, and derives multiple new local and global consequences in spectral and extremal graph theory.

Findings

Extended spectral Turán inequality to weighted graphs.

Characterized extremal graphs attaining the bound.

Derived new local and global spectral and extremal graph theorems.

Abstract

Confirming a conjecture of Elphick and Edwards and strengthening a spectral theorem of Wilf, Nikiforov proved that for any $K_{r+1}$-free graph $G$, $\lambda(G)^2 \leq 2 (1 - 1/r) m$, where $\lambda(G)$ is the spectral radius of $G$, and $m$ is the number of edges of $G$. This result was later improved in \cite{LiuN26}, where it was shown that for any graph $G$, $\lambda(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\mathrm{cl}(e) - 1}{\mathrm{cl}(e)}$, where $\mathrm{cl}(e)$ denotes the order of the largest clique containing the edge $e$. In this paper, we further extend this inequality to weighted graphs, proving that \[ \lambda(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\mathrm{cl}(e) - 1}{\mathrm{cl}(e)} w(e)^2, \] and we characterize all extremal graphs attaining this bound. Our main theorem yields several new consequences, including two vertex-based and vertex-degree-based local versions of Tur\'an's theorem, as well as weighted generalizations of the Edwards--Elphick theorem and the Cvetkovi\'c theorem, and two localized versions of Wilf's theorems. One of these localized Wilf's theorem confirms a conjecture that originates from Probability and Operator Algebras and was proposed by R. Tripathi independently of us. Moreover, our main result unifies and implies numerous earlier ones from spectral graph theory and extremal graph theory, including Stanley's spectral inequality, Hong's inequality, a localized Tur\'an-type theorem, and a recent extremal theorem by Adak and Chandran. Notably, while Nikiforov's earlier spectral inequality implied Stanley's bound, it did not imply Hong's inequality -- a gap that is now bridged by our result. As a key tool, we establish the inequality $\sum_{e \in E(G)} \frac{2}{\mathrm{cl}(e)} \geq n-1$, which complements an upper bound $\sum_{e \in E(G)} \frac{2}{\mathrm{cl}(e)-1} \leq n^2 - 2m$ due to Brada\v{c}, and Malec and Tompkins, independently.