The nullity of a graph G is the multiplicity of zero as an eigenvalue of its adjacency matrix. An assignment of weights to the vertices of a graph, that satisfies a zero sum condition over the neighbors of each vertex, and uses maximum number of independent variables is denoted by a high zero sum weighting of the graph. This applicable tool is used to determine the nullity of the graph. Two types of graphs are defined, and the change of their nullities is studied, namely, the graph G+ab constructed from G by adding a new vertex ab which is joint to all neighbors of both vertices a and b of G, and G•ab which is obtained from G+ab by removing both vertices a and b.

In geographic, the eigenvalues and eigenvectors of transportation network provides many informations about its connectedness. It is proven that the more highly connected in a transportation network G has largest eigenvalue and hence more multiple occurrences of the eigenvalue -1. For a graph G with adjacency matrix A, the multiplicity of the eigenvalue -1 equals the dimension of the null space of the matrix A + I. In this paper, we constructed a high closed zero sum weighting of G and by which its proved that, the dimension of the null space of the eigenvalue -1 is the same as the number of independent variables used in a non-trivial high closed zero sum weighting of the graph. Multiplicity of -1 as an eigenvalue of known graphs and of corona product of certain classes of graphs are determined and two classes of -1- nut graphs are constructed.

For the adjacency matrix A of a graph G, a number  is an eigenvalue of G if for some non zerovector X, AX=X. The vector X is called the eigenvector corresponding to . The eigenvalues are exactly those numbers  that make the matrix A- I to be singular. All eigenvectors corresponding to  forms a subspace V; the dimension of V is equal to the multiplicity of . A graph G is a Smith graph if 2 is an eigenvalue of the adjacency matrix A of G, a -weighting technique is introduced and applied to characterize some classes of Smith graphs as well as to study their nullities and the nullity of vertex identification of such graphs. We also have proved that under certain conditions the vertex identification of some Smith graphs is a Smith graph.