On the permanental nullity and matching number of graphs

TL;DR

This paper explores the relationship between the permanental nullity and matching number of graphs, providing formulas, conditions, and bounds for various classes of graphs using the Gallai-Edmonds structure theorem.

Contribution

It introduces a concise formula linking permanental nullity and matching number, and characterizes graphs with nullity zero, applying these results to specific graph classes.

Findings

Derived a formula relating permanental nullity and matching number.

Established necessary and sufficient conditions for nullity zero.

Provided bounds for unicyclic graphs' nullity and nullity values for line and factor critical graphs.

Abstract

For a graph $G$ with $n$ vertices, let $\nu(G)$ and $A(G)$ denote the matching number and adjacency matrix of $G$, respectively. The permanental polynomial of $G$ is defined as $\pi(G,x)={\rm per}(Ix-A(G))$. The permanental nullity of $G$, denoted by $\eta_{per}(G)$, is the multiplicity of the zero root of $\pi(G,x)$. In this paper, we use the Gallai-Edmonds structure theorem to derive a concise formula which reveals the relationship between the permanental nullity and the matching number of a graph. Furthermore, we prove a necessary and sufficient condition for a graph $G$ to have $\eta_{per}(G)=0$. As applications, we show that every unicyclic graph $G$ on $n$ vertices satisfies $n-2\nu(G)-1 \le \eta_{per}(G) \le n-2\nu(G)$, that the permanental nullity of the line graph of a graph is either zero or one, and that the permanental nullity of a factor critical graph is always zero.