Localization of spectral Turán-type theorems
M. Rajesh Kannan, Hitesh Kumar, Shivaramakrishna Pragada
TL;DR
The paper develops a program to localize spectral Turán-type inequalities by substituting global graph parameters with local clique metrics c(v) and c(e). It derives vertex- and edge-localized bounds, proves them in diamond-free and random graphs, and extends Erdős–Stone–Simonovits-type results by replacing spectral radii with sqrt{s^{+}} in a localized setting. A key technical tool is a triangle-count–based bound on sqrt{s^{+}}, together with weighted Motzkin–Straus-type results and majorization; these enable localized Wilf-type and Bollobás–Nikiforov-type inequalities and a localized Nikiforov walk inequality. Collectively, the results advance a broader program of localizing spectral extremal inequalities, with implications for sparse and structured graph classes such as diamond-free and F-free graphs, as well as random graphs. The work also provides concrete conjectures and proofs in several nontrivial cases and highlights avenues for further refinement via local parameters in spectral graph theory.
Abstract
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 $λ_1 \ge \cdots \ge λ_n$. We study localized refinements of spectral Turán-type theorems by replacing global parameters such as the clique number $ω(G)$, 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 $s^+(G) = \sum_{λ_i > 0}λ_i^2$. Inspired by the Bollobás-Nikiforov conjecture (2007) on the first two eigenvalues, we then introduce an edge-localized analogue: \[λ_1^2(G) + λ_2^2(G) \le \sum_{e\in E(G)} 2\left(1-\frac{1}{c(e)}\right).\] 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 Erdős, Stone and Simonovits Theorem by replacing the spectral radius with $\sqrt{s^{+}(G)}$ and establish it for all $F$-free graphs with $χ(F)=3$. 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.