# The inertia set of a signed graph

Research paper by **Marina Arav, Frank J. Hall, Zhongshan Li, Hein van der Holst**

Indexed on: **26 Aug '12**Published on: **26 Aug '12**Published in: **Mathematics - Combinatorics**

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#### 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.