2026/07/20 by Yinchen Liu, Quanyu Tang, Shengtong Zhang · 1 citation
#math.CO
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)\≥ 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.