Symmetry and asymmetry between positive and negative square energies of graphs

TL;DR

This paper explores the relationship between positive and negative square energies of graphs, highlighting instances of symmetry and asymmetry, and addressing open questions in the spectral graph theory domain.

Contribution

It reviews known symmetry and asymmetry in graph square energies, provides new examples, and proposes future research directions.

Findings

Identified asymmetry in large random graphs

Provided new examples of symmetry in square energies

Addressed open questions about positive and negative energies

Abstract

The positive and negative square energies of a graph, $s^+(G)$ and $s^-(G)$, are the sums of squares of the positive and negative eigenvalues of the adjacency matrix, respectively. The first results on square energies revealed symmetry between $s^+(G)$ and $s^-(G)$. This paper reviews examples of asymmetry between these parameters, for example using large random graphs and the ratios $s^+/s^-$ and $s^-/s^+$, as well as new examples of symmetry. We answer some questions previously asked about $s^{+}$ and $s^{-}$ and suggest several further avenues of research.