On the nullity of tricyclic graphs

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G)⩽n-2 if G is a simple graph on n vertices and G is not isomorphic to nK1. The extremal graphs attaining the upper bound n-2 and the second upper bound n-3 have been obtained. In this paper, the graphs with nullity n-4 are characterized. Furthermore the tricyclic graphs with maximum nullity are discussed.