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.

Disclosure

“Now s+ (Kn ) = (n − 1)2 , s− (Kn ) = n − 1, whereas both square energies of an edgeless graph vanish. The equality statements follow. □ Acknowledgments and AI disclosure We used ChatGPT to generate exploratory code for testing conjectures and to help polish the language of proof drafts. All mathematical content was verified by the authors, and the proof of Theorem 1.2 has additionally been formally verified in the Lean pr”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T18:59:57.047857Z

Structural counts