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

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.