On the Laplacian eigenvalues of a graph and Laplacian energy

Let G be a simple graph with n vertices, m   edges, maximum degree Δ, average degree d‾=2mn, clique number ω   having Laplacian eigenvalues μ1,μ2,…,μn−1,μn=0μ1,μ2,…,μn−1,μn=0. For k   (1≤k≤n1≤k≤n), let Sk(G)=∑i=1kμi and let σ   (1≤σ≤n−11≤σ≤n−1) be the number of Laplacian eigenvalues greater than or equal to average degree d‾. In this paper, we obtain a lower bound for Sω−1(G)Sω−1(G) and an upper bound for Sσ(G)Sσ(G) in terms of m, Δ, σ and clique number ω   of the graph. As an application, we obtain the stronger bounds for the Laplacian energy LE(G)=∑i=1n|μi−d‾|, which improve some well known earlier bounds.