On the symmetry of the Laplacian spectra of signed graphs

TL;DR

This paper introduces a new method to generate symmetric spectra for signed graphs, linking spectral symmetry to graph properties and heat equation solutions, extending bipartiteness concepts.

Contribution

It develops a novel machinery for symmetric spectra in signed graphs and establishes a fundamental connection with damped two-periodic solutions of the heat equation.

Findings

New machinery for symmetric spectra in signed graphs

Connection between spectral symmetry and heat equation solutions

Includes bipartiteness as a special case

Abstract

We study the symmetry properties of the spectra of normalized Laplacians on signed graphs. We find a new machinery that generates symmetric spectra for signed graphs, which includes bipartiteness of unsigned graphs as a special case. Moreover, we prove a fundamental connection between the symmetry of the spectrum and the existence of damped two-periodic solutions for the discrete-time heat equation on the graph.