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.