On Laplacian energy, Laplacian-energy-like invariant and Kirchhoff index of graphs

Let G be a connected graph of order n and size m with Laplacian eigenvalues μ1≥μ2≥⋯≥μn=0. The Kirchhoff index of G, denoted by Kf, is defined as: Kf=n∑i=1n−11μi. The Laplacian-energy-like invariant (LEL) and the Laplacian energy (LE) of the graph G, are defined as: LEL=∑i=1n−1μi and LE=∑i=1n|μi−2mn|, respectively. We obtain two relations on LEL with Kf, and LE with Kf. For two classes of graphs, we prove that LEL>Kf. Finally, we present an upper bound on the ratio LE/LEL and characterize the extremal graphs.