Signed graphs with exactly two main eigenvalues: The unicyclic case

TL;DR

This paper characterizes signed graphs with exactly two main eigenvalues when their associated multigraph has a unicyclic base, extending previous work from tree-based to unicyclic-based structures.

Contribution

It extends the classification of signed graphs with two main eigenvalues to those with unicyclic base graphs, building on prior tree-based analyses.

Findings

Characterization of signed graphs with two main eigenvalues in the unicyclic case

Identification of structural properties of such graphs

Open problems proposed for future research

Abstract

An eigenvalue $\lambda$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Du et al. (2024, 2026) characterized a class of signed graphs with exactly two main eigenvalues by analyzing a type of multigraph whose base graph is a tree. In this paper, we extend this study to the case where the associated multigraph has a unicyclic base graph, and we conclude by proposing several open problems.