The Equality Case in the Positive Square-Energy Strengthening of Turán's Theorem
Let $G$ be a graph of order $n$ with eigenvalues $λ_1(G) \geq \dots \geq λ_n(G)$, and let $s_+(G)=\sum_{λ_i(G)>0}λ_i(G)^2.$ Recently Liu, Tang, and Zhang proved the positive square-energy strengthening of Turán's theorem \[\sqrt{s_+(G)}\leq \left(1-\frac1r\right)n.\] where $r=ω(G)$ is the clique number of $G$. We characterize the...