The nullity of (k-1)-cyclic graphs

A simple connected graph is called a k-cyclic graph provided that the number of edges equals its order plus (k-1). In this paper, we define a class of (k-1)-cyclic graph, denoted by Θk. The singularity of Θ-graph is studied. Also, the (k-1)-cyclic graphs that contain Θ-graph as an induced subgraph are paid much attention. We characterize those graphs that have extreme nullities. The results generalize the corresponding theorems on unicyclic and bicyclic graphs.