Open problems in the spectral theory of signed graphs

TL;DR

This paper surveys the spectral theory of signed graphs, highlighting recent interest, generalizations from unsigned graphs, and open problems in understanding their adjacency spectra.

Contribution

It provides an overview of key results and outlines open problems in the spectral analysis of signed graphs, emphasizing their connection to unsigned graph theory.

Findings

Spectral theory of signed graphs generalizes unsigned graph spectra.

Balanced signed graphs relate to unsigned graphs.

Open problems remain in understanding spectral properties of signed graphs.

Abstract

Signed graphs are graphs whose edges get a sign $+1$ or $-1$ (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs. Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs.