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In this paper, a high zero-sum weighting is applied to evaluate the nullity of a dendrimer graph for some special graphs such as cycles, paths, complete graphs, complete bipartite graphs and star graphs.Finally, we introduce and prove a sharp lower and a sharp upper bound for the nullity of the coalescence graph of two graphs.

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.

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