A new spectral Turán theorem for weighted graphs and consequences

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 localized versions of Wilf's two theorems. Moreover, the 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.