On the nullity of a graph with cut-points

Let G be a simple graph of order n and A(G) be its adjacency matrix. The nullity of a graph C. denoted by eta(G). is the multiplicity of the eigenvalue zero in the spectrum of A(G). Denote by C-k and L-k, the set of all connected graphs with k induced cycles and the set of line graphs of all graphs in C-k, 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 L-0 is {0,1}. 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 G is an element of C-k and eta(L(G)) = k + 1, and the nullity set of L-k is {0, 1,..., k k + 1} for any given k, where L(G) denotes the line graph of the graph G. (C) 2011 Elsevier Inc. All rights reserved.