A positive square-energy strengthening of Turán's theorem

Abstract

Let $G$ be an $n$-vertex graph with clique number $ω(G)$, and let $s^+(G)$ denote the sum of the squared positive adjacency eigenvalues. We prove that $$ \sqrt{s^+(G)}\le\left(1-\frac{1}{ω(G)}\right)n. $$ This strengthens Wilf's classical spectral Turán 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.

Disclosure

“≤ ω, proving n− s (G) Conjecture 1.1. Acknowledgments and AI disclosure We used ChatGPT to generate exploratory code for testing conjectures and to help polish the language of proof drafts. All mathematical content was verified by the authors. The core ideas and the overall strategy of the proof were developed independently b”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T18:58:45.093275Z

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