On the nullity and the matching number of unicyclic graphs

Abstract

Let G be a graph with n vertices and ν ( G ) be the matching number of G . Let η ( G ) denote the nullity of G (the multiplicity of the eigenvalue zero of G ). It is well known that if G is a tree, then η ( G ) = n - 2 ν ( G ) . Tan and Liu [X. Tan, B. Liu, On the nullity of unicyclic graphs, Linear Alg. Appl. 408 (2005) 212–220] proved that the nullity set of all unicyclic graphs with n vertices is { 0 , 1 , … , n - 4 } and characterized the unicyclic graphs with η ( G ) = n - 4 . In this paper, we characterize the unicyclic graphs with η ( G ) = n - 5 , and we prove that if G is a unicyclic graph, then η ( G ) equals n - 2 ν ( G ) - 1 , n - 2 ν ( G ) , or n - 2 ν ( G ) + 2 . We also give a characterization of these three types of graphs. Furthermore, we determine the unicyclic graphs G with η ( G ) = 0 , which answers affirmatively an open problem by Tan and Liu.