On the permanental nullity and matching number of graphs

Abstract

For a graph G with n vertices, let and A(G) denote the matching number and adjacency matrix of G, respectively. The permanental polynomial of G is defined as . The permanental nullity of G, denoted by , is the multiplicity of the zero root of . 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 . As applications, we show that every unicyclic graph G on n vertices satisfies , 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.