# Two-connected signed graphs with maximum nullity at most two

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

Indexed on: **09 Jul '14**Published on: **09 Jul '14**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 $V=\{1,\ldots,n\}$ and
$\Sigma\subseteq E$. The edges in $\Sigma$ are called odd and the other edges
of $E$ even. By $S(G,\Sigma)$ we denote the set of all symmetric $n\times n$
matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are adjacent and
connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are adjacent and
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
parameters $M(G,\Sigma)$ and $\xi(G,\Sigma)$ of a signed graph $(G,\Sigma)$ are
the largest nullity of any matrix $A\in S(G,\Sigma)$ and the largest nullity of
any matrix $A\in S(G,\Sigma)$ that has the Strong Arnold Hypothesis,
respectively. In a previous paper, we gave a characterization of signed graphs
$(G,\Sigma)$ with $M(G,\Sigma)\leq 1$ and of signed graphs with
$\xi(G,\Sigma)\leq 1$. In this paper, we characterize the $2$-connected signed
graphs $(G,\Sigma)$ with $M(G,\Sigma)\leq 2$ and the $2$-connected signed
graphs $(G,\Sigma)$ with $\xi(G,\Sigma)\leq 2$.