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 μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n = 0 . The Kirchhoff index of G , denoted by Kf , is defined as: K f = n ∑ i = 1 n − 1 1 μ i . The Laplacian-energy-like invariant ( L E L ) and the Laplacian energy ( L E ) of the graph G , are defined as: L E L = ∑ i = 1 n − 1 μ i and L E = ∑ i = 1 n | μ i − 2 m n | , respectively. We obtain two relations on LEL with Kf , and LE with Kf . For two classes of graphs, we prove that L E L > K f . Finally, we present an upper bound on the ratio L E / L E L and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]