The Nullity of Bicyclic Graphs in Terms of Their Matching Number

Abstract:

      Let $G$ be a graph with $n(G)$ vertices and $m( G )$ be its matching number. The nullity of $G$, denoted by $\eta( G )$, is the multiplicity of the eigenvalue zero of adjacency matrix of $G$. It is well known that if $G$ is a tree, then $\eta( G )=n( G )-2 m( G )$. Guo et al. [Jiming GUO, Weigen YAN, Yeongnan YEH. On the nullity and the matching number of unicyclic graphs. Linear Alg. Appl., 2009, 431: 1293--1301] proved that if $G$ is a unicyclic graph, then $\eta( G )$ equals $n( G ) - 2 m( G ) -1$, $n( G ) - 2 m( G ) $, or $n( G ) - 2 m( G ) +2$. In this paper, we prove that if $G$ is a bicyclic graph, then $\eta( G )$ equals $n( G ) - 2 m( G ) $, $n (G)-2m(G)\pm 1 $, $n( G ) - 2 m( G )\pm 2$ or $n( G ) - 2 m (G ) +4$. We also give a characterization of these six types of bicyclic graphs corresponding to each nullity.