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arXiv · 2608.24861

A Vertex-Localized Positive Square-Energy Strengthening of Tur\'an's Theorem

Abhay Jayarajan·M. Rajesh Kannan·Shivaramakrishna Pragada·Rahul Roy
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Abstract

Let $G$ be a graph of order $n$ with the adjacency eigenvalues $\lambda_1(G) \geq \dots \geq \lambda_n(G) $. Let $c(v)$ denote the maximum order of a clique containing vertex $v$. We prove the vertex-localized positive square-energy inequality \[ \sqrt{s_+(G)} \leq \sum_{v\in V}\left(1-\frac1{c(v)}\right), \] where \[ s_+(G)=\sum_{\lambda_i(G)>0}\lambda_i(G)^2. \] We also characterize equality. Apart from edgeless graphs, equality holds precisely for graphs obtained from a complete regular multipartite graph by adding an arbitrary number of isolated vertices. This settles a conjecture of Kannan, Kumar and Pragada.

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Abhay Jayarajan, M. Rajesh Kannan, Shivaramakrishna Pragada, Rahul Roy. 2026-08-25. A Vertex-Localized Positive Square-Energy Strengthening of Tur\'an's Theorem. https://arxiv.org/abs/2608.24861

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