Paper Title
K-Closed Nbd and K-Open Nbd Matrices of Graphs and Their Energy

Abstract
From a distance matrix of a graph one can determine the distance from one vertex to another vertex. In order to find how may vertices are at a distance k from a given vertex, we introduce two new binary matrices using the distance concept in graph. First we define the k-closed nbd matrix and k-open nbd matrix of a graph. Then we find some relation between these matrices and adjacency matrix. Like adjacency matrix, the rows and columns of these matrices correspond to an arbitrary labelling of the vertices of the graph. Hence we shall be interested primarily in those properties of these matrices which are invariant under permutations of the rows and columns. Foremost among such properties are the spectral properties of the matrix. So we then focused on characteristic polynomial, spectrum and energy of these matrices, namely k-closed nbd polynomial, k-closed nbd spectrum, k-closed nbd energy, k-open nbd polynomial, k-open nbd spectrum, and k-open nbd energy. Finally, we illustrate all the new definitions we made on the molecular graph of Isopentane. Keywords - k- closed nbd matrix, k- closed nbd polynomial, k- closed nbd spectrum, k- closed nbd energy, k-open nbd matrix, k-open nbd polynomial,k-open nbd spectrum, k-open nbd energy.