two laplacian energies and the relations between them

Introduction/purpose: the laplacian energy (le) is the sum of absolute values of the terms μi-2m/n, where μi, i=1,2,…,n, are the eigenvalues of the laplacian matrix of the graph g with n vertices and m edges. in 2006, another quantity z was introduced, based on laplacian eigenvalues, which was also named laplacian energy“. z is the sum of squares of laplacian eigenvalues. the aim of this work is to establish relations between le and z. results: lower and upper bounds for le are deduced, in terms of z. conclusion: the paper contributes to the laplacian spectral theory and the theory of graph energies. it is shown that, as a rough approximation, le is proportional to the tem (z-4m²/n)^1/2.