Some signed graphs whose eigenvalues are main

TL;DR

This paper investigates conditions under which all eigenvalues of certain signed graphs are main, proving the existence of switchings that achieve this for specific graph classes, thus supporting a conjecture in spectral graph theory.

Contribution

It demonstrates that for complete multipartite graphs, harmonic trees, and graphs of the form S_{n,k}, there exists a switching making all eigenvalues main, confirming a conjecture by Akbari et al.

Findings

Existence of switchings making all eigenvalues main for specific graph classes.

Supports a conjecture by Akbari et al. on eigenvalue properties of signed graphs.

Provides spectral characterizations for classes of signed graphs.

Abstract

Let $G$ be a graph. For a subset $X$ of $V(G)$, the switching $\sigma$ of $G$ is the signed graph $G^{\sigma}$ obtained from $G$ by reversing the signs of all edges between $X$ and $V(G)\setminus X$. Let $A(G^{\sigma})$ be the adjacency matrix of $G^{\sigma}$. An eigenvalue of $A(G^{\sigma})$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Let $S_{n,k}$ be the graph obtained from the complete graph $K_{n-r}$ by attaching $r$ pendent edges at some vertex of $K_{n-r}$. In this paper we prove that there exists a switching $\sigma$ such that all eigenvalues of $G^{\sigma}$ are main when $G$ is a complete multipartite graph, or $G$ is a harmonic tree, or $G$ is $S_{n,k}$. These results partly confirm a conjecture of Akbari et al.