The positive and negative square-energy conjecture

Abstract

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.

12 pages. Comments and suggestions are welcome