Localization of spectral Tur\'{a}n-type theorems

Let $G$ be a graph, and let $v$ and $e$ be a vertex and an edge of $G$, respectively. Define $c(v)$ (resp. $c(e)$) to be the order of the largest clique in $G$ containing $v$ (resp. $e$). Denote the adjacency eigenvalues of $G$ by $\lambda_1 \ge \cdots \ge \lambda_n$. We study localized refinements of spectral Tur\'{a}n-type theorems by replacing global parameters such as the clique number $\omega(G)$, size $m$ and order $n$ of $G$ with local quantities $c(v)$ and $c(e)$. Motivated by a conjecture of Elphick, Linz and Wocjan (2024), we first propose a vertex-localized strengthening of Wilf's inequality: $$\sqrt{s^{+}(G)} \le \sum_{v\in V(G)}\left(1-\frac{1}{c(v)}\right),$$ where $s^+(G) = \sum_{\lambda_i > 0}\lambda_i^2$. Inspired by the Bollob\'{a}s-Nikiforov conjecture (2007) on the first two eigenvalues, we then introduce an edge-localized analogue: $$\lambda_1^2(G) + \lambda_2^2(G) \le \sum_{e\in E(G)} 2\left(1-\frac{1}{c(e)}\right).$$ As evidence of their validity, we verify the above conjectures for diamond-free graphs and random graphs. We also propose strengthening of the spectral versions of the Erd\H{o}s, Stone and Simonovits Theorem by replacing the spectral radius with $\sqrt{s^{+}(G)}$ and establish it for all $F$-free graphs with $\chi(F)=3$. A key ingredient in our proofs is a general upper bound relating $\sqrt{s^{+}(G)}$ to the triangle count $t(G)$. Finally, we prove a localized version of Nikiforov's walk inequality and conjecture a stronger localized version. These results contribute to the broader program of localizing spectral extremal inequalities.