Bounds for laplacian-type graph energies

© 2015 Miskolc University Press. Let G be an undirected simple and connected graph with n vertices (n ≥ 3) and m edges. Denote by μ1 ≥ μ2 ≥ ... ≥ μn-1 > μn = 0, γ1 ≥ γ2 ≥ ... ≥ γn, and ρ1 ≥ ρ2 ≥ ... ≥ ρn-1 > ρn = 0, respectively, the Laplacian, signless Laplacian, and normalized Laplacian eigenvalues of G. The Laplacian energy, signless Laplacian energy, and normalized Laplacian energy of G are defined as LE = Σni=1 |μi-2m/n|, SLE = Σni=1 |γi-2m/n|, and NLE = Σni=1 |ρi-1|, respectively. Lower bounds for LE, SLE, and NLE are obtained