Two-connected signed graphs with maximum nullity at most two

TL;DR

This paper characterizes 2-connected signed graphs with maximum nullity at most two, extending previous work on graphs with nullity at most one, and provides a structural understanding of such graphs.

Contribution

It offers a complete characterization of 2-connected signed graphs with nullity at most two, advancing the understanding of their structural properties.

Findings

Characterization of 2-connected signed graphs with nullity ≤ 2

Extension of previous nullity ≤ 1 results to nullity ≤ 2

Provides structural criteria for these 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 $V=\{1,\ldots,n\}$ 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 $\xi(G,\Sigma)\leq 2$.