The Equality Case in the Positive Square-Energy Strengthening of Tur\'an's Theorem

Abstract

Let $G$ be a graph of order $n$ with eigenvalues $\lambda_1(G) \geq \dots \geq \lambda_n(G)$, and let $s_+(G)=\sum_{\lambda_i(G)>0}\lambda_i(G)^2.$ Recently Liu, Tang, and Zhang proved the positive square-energy strengthening of Tur\'an's theorem \[\sqrt{s_+(G)}\leq \left(1-\frac1r\right)n.\] where $r=\omega(G)$ is the clique number of $G$. We characterize the families of graphs for which the above inequality is sharp. Precisely, we prove that, for $r\geq 2$, equality holds if and only if $r\mid n$ and $G$ is the complete regular $r$-partite graph $K_{n/r,\ldots,n/r}$.