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≥⋯≥μ_n=0. The Kirchhoff index of G, denoted by Kf, is defined as: Kf=n∑_i=1 ⁿ⁻¹[Formula presented]. The Laplacian-energy-like invariant (LEL) and the Laplacian energy (LE) of the graph G, are defined as: LEL=∑_i=1 ⁿ⁻¹μ_i and LE=∑i=1n|μ_i−[Formula presented]|, 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.