The inertia set of a signed graph

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.