A positive square-energy strengthening of Turán's theorem

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 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.

11 pages. Comments and suggestions are welcome