The positive and negative square-energy conjecture

Let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of a graph $G$, respectively. We prove the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies $$\min\{s^+(G),s^-(G)\}\ge n-1.$$ The proof introduces a new framework for square-energy estimates, in which the Hadamard squares of positive semidefinite matrices that encode these spectral quantities are relaxed to the full doubly nonnegative cone.