On the nullity of a graph with cut-points()

Let G be a simple graph of order n and [Formula: see text] be its adjacency matrix. The nullity of a graph G, denoted by [Formula: see text] , is the multiplicity of the eigenvalue zero in the spectrum of [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the set of all connected graphs with k induced cycles and the set of line graphs of all graphs in [Formula: see text] , respectively. In 1998, Sciriha [I. Sciriha, On singular line graphs of trees, Congr. Numer. 135 (1998) 73–91] show that the order of every tree whose line graph is singular is even. Then Gutman and Sciriha [I. Gutman, I. Sciriha, On the nullity of line graphs of trees, Discrete Math. 232 (2001) 35–45] show that the nullity set of [Formula: see text] is [Formula: see text]. In this paper, we investigate the nullity of graphs with cut-points and deduce some concise formulas. Then we generalize Scirihas’ result, showing that the order of every graph G is even if such a graph G satisfies that [Formula: see text] and [Formula: see text] , and the nullity set of [Formula: see text] is [Formula: see text] for any given k, where [Formula: see text] denotes the line graph of the graph G.