The Nullity of Bicyclic Graphs in Terms of Their Matching Number

Let G be a graph with n(G) vertices and m(G) be its matching number. The nullity of G, denoted by η(G), is the multiplicity of the eigenvalue zero of adjacency matrix of G. It is well known that if G is a tree, then η(G) = n(G)− 2m(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 η(G) equals n(G)−2m(G)−1, n(G)−2m(G), or n(G)−2m(G)+2. In this paper, we prove that if G is a bicyclic graph, then η(G) equals n(G)− 2m(G), n(G)− 2m(G)± 1, n(G)− 2m(G)± 2 or n(G) − 2m(G) + 4. We also give a characterization of these six types of bicyclic graphs corresponding to each nullity.