Remark on the Laplacian-energy-like and Laplacian incidence energy invariants of graphs

Let GG be an undirected connected graph with nn vertices and mm edges, n≥3n≥3, and let μ1≥μ2≥⋯≥μn−1>μn=0μ1≥μ2≥⋯≥μn−1>μn=0 and ρ1≥ρ2≥⋯≥ρn−1>ρn=0ρ1≥ρ2≥⋯≥ρn−1>ρn=0 be Laplacian and normalized Laplacian eigenvalues of GG, respectively. The Laplacian-energy-like (LEL) invariant of graph GG is defined as LEL(G)=∑n−1i=1μi−−√LEL(G)=∑i=1n−1μi. The Laplacian incidence energy of graph is defined as LIE(G)=∑n−1i=1ρi−−√LIE(G)=∑i=1n−1ρi. In this paper, we consider lower bounds of graph invariants LEL(G)LEL(G) and LIE(G)LIE(G) in terms of some graph parameters, that refine some known results.