Extremal values for the square energies of graphs

Let $G$ be a graph with $n$ non-isolated vertices and $m$ edges. The positive / negative square energies of $G$, denoted $s^+(G)$ / $s^-(G)$, are defined as the sum of squares of the positive / negative eigenvalues of the adjacency matrix $A_G$ of $G$. In this work, we provide several new tools for studying square energy encompassing semi-definite optimization, graph operations, and surplus. Using our tools, we prove the following results on the extremal values of $s^{\pm}(G)$ with a given number of vertices and edges. 1. We have $\min(s^+(G), s^-(G)) \geq n - \gamma \geq \frac{n}{2}$, where $\gamma$ is the domination number of $G$. This verifies a conjecture of Elphick, Farber, Goldberg and Wocjan up to a constant, and proves a weaker version of this conjecture introduced by Elphick and Linz. 2. We have $s^+(G) \geq m^{6/7 - o(1)}$ and $s^-(G) = \Omega(m^{1/2})$, with both exponents being optimal.