The signed graphs with symmetric spectra

TL;DR

This paper characterizes signed graphs with symmetric spectra, providing conditions and methods for constructing such graphs, including infinite families, extending beyond bipartite graphs.

Contribution

It offers new necessary and sufficient conditions for spectrally symmetric signed graphs and introduces methods to construct infinite families of these graphs.

Findings

Identifies conditions for spectral symmetry in signed graphs.

Provides methods to construct signed graphs with symmetric spectra.

Produces infinite families of spectrally symmetric signed graphs.

Abstract

It is well known that a graph $G$ has a symmetric spectrum if and only if it is bipartite, a signed graph $\Gamma=(G,\sigma)$ has a symmetric spectrum if $G$ is bipartite. However, there exists a spectrally symmetric signed graph $\Gamma=(G,\sigma)$ such that $G$ is not bipartite. In this paper, we focus to characterize the signed graphs with symmetric spectra. Some necessary and (or) sufficient conditions for spectrally symmetric signed graphs are given. Moreover, some methods to construct signed graphs with symmetric spectra are found and infinite families of these signed graphs are produced.