On Laplacian energy in terms of graph invariants

For G being a graph with n vertices and m   edges, and with Laplacian eigenvalues μ1≥μ2≥⋯≥μn−1≥μn=0,μ1≥μ2≥⋯≥μn−1≥μn=0, the Laplacian energy is defined as LE=∑i=1n|μi−2m/n|. Let σ be the largest positive integer such that μσ ≥ 2m/n  . We characterize the graphs satisfying σ=n−1σ=n−1. Using this, we obtain lower bounds for LE in terms of n, m, and the first Zagreb index. In addition, we present some upper bounds for LE in terms of graph invariants such as n, m, maximum degree, vertex cover number, and spanning tree packing number.