The Equality Case in the Positive Square-Energy Strengthening of Turán's Theorem

Let $G$ be a graph of order $n$ with eigenvalues ${\lambda }_1(G)\ge \cdots \ge {\lambda }_n(G)$, and let $s_{+}(G)={\sum }_{{\lambda }_i(G)>0}{\lambda }_i(G{)}^2\mathrm{.}$ Recently Liu, Tang, and Zhang proved the positive square-energy strengthening of Turán’s theorem $$\sqrt{s_{+}(G)}\le (1-\frac{1}{r})n\mathrm{.}$$ 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 $r\ge 2$, equality holds if and only if $r\mid n$ and $G$ is the complete regular $r$-partite graph $K_{n\mathrm{/}r,\dots ,n\mathrm{/}r}$.