The Nullity of Bicyclic Graphs in Terms of Their Matching Number 
Received:October 10, 2015 Revised:October 12, 2016 
Key Words:
nullity bicyclic graphs matching number

Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11331003; 11471077). 

Hits: 2053 
Download times: 1965 
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: 12931301] 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. 
Citation: 
DOI:10.3770/j.issn:20952651.2016.06.001 
View Full Text View/Add Comment Download reader 


