On energy of matrices of a weighted graph

Abstract

The energy of a square matrix is the sum of the absolute values of deviations of its eigenvalues from their mean. In this paper, we extend the notion of the Laplacian and the signless Laplacian energy of a graph to non-negative real symmetric matrices with zero diagonal. Such a matrix [Formula: see text] can be considered as the adjacency matrix of some weighted graph. Considering the sum of the weights of edges that are incident to a vertex as the weight of the corresponding vertex, we construct the diagonal matrix [Formula: see text] of which the diagonal entries are the weights of the corresponding vertices. We define a weighted version of the Laplacian matrix as [Formula: see text] and that of the signless Laplacian matrix as [Formula: see text]. In this work, we obtain some bounds of the energies of the weighted adjacency matrix, the weighted Laplacian matrix, and the weighted signless Laplacian matrix by studying the corresponding weighted graph. We also obtain some relations interconnecting those energies.