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arXiv · 2607.18031

The positive and negative square-energy conjecture

Yinchen Liu·Quanyu Tang·Shengtong Zhang
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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.

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Yinchen Liu, Quanyu Tang, Shengtong Zhang. 2026-07-20. The positive and negative square-energy conjecture. https://arxiv.org/abs/2607.18031

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