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arXiv · 2607.18044

A positive square-energy strengthening of Tur\'an's theorem

Yinchen Liu·Quanyu Tang·Shengtong Zhang
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Abstract

Let $G$ be an $n$-vertex graph with clique number $\omega(G)$, and let $s^+(G)$ denote the sum of the squared positive adjacency eigenvalues. We prove that $$ \sqrt{s^+(G)}\le\left(1-\frac{1}{\omega(G)}\right)n. $$ This strengthens Wilf's classical spectral Tur\'{a}n theorem and resolves a conjecture of Elphick and Wocjan. Adopting the relaxation of our companion paper on the square-energy conjecture, we reduce the theorem to a Motzkin--Straus inequality for doubly nonnegative matrices, which we prove via a local inverse-probability estimate for the Caro--Wei greedy algorithm on the complement.

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Yinchen Liu, Quanyu Tang, Shengtong Zhang. 2026-07-20. A positive square-energy strengthening of Tur\'an's theorem. https://arxiv.org/abs/2607.18044

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