The inertia set of a signed graph

TL;DR

This paper investigates the stable inertia set of signed graphs, characterizing the eigenvalue signatures of matrices associated with these graphs that satisfy the Strong Arnold Property.

Contribution

It introduces the concept of the stable inertia set for signed graphs and analyzes its properties, providing new insights into the spectral structure of signed graphs.

Findings

Characterization of the stable inertia set for signed graphs

Identification of conditions for matrices with the Strong Arnold Property

Insights into the spectral properties of signed graphs

Abstract

A signed graph is a pair $(G,\Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V={1,...,n}$ and $\Sigma\subseteq E$. By $S(G,\Sigma)$ we denote the set of all symmetric $V\times V$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are connected by only odd edges, $a_{i,j}\in \mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i\not=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} \in \mathbb{R}$ for all vertices $i$. The stable inertia set of a signed graph $(G,\Sigma)$ is the set of all pairs $(p,q)$ for which there exists a matrix $A\in S(G,\Sigma)$ with $p$ positive and $q$ negative eigenvalues which has the Strong Arnold Property. In this paper, we study the stable inertia set of (signed) graphs.