On the bounds for signless Laplacian energy of a graph

Abstract

For a simple graph G with n-vertices, m edges and having signless Laplacian eigenvalues q1,q2,…,qn, the signless Laplacian energy QE(G) of the graph G is defined as QE(G)=∑i=1n∣qi−d¯∣, where d¯=2mn is the average degree of G. In this paper, we obtain the lower and upper bounds for the signless Laplacian energy QE(G) in terms of clique number ω, maximum degree Δ, number of vertices n, first Zagreb index M1(G) and number of edges m. As an application, we obtain the bounds for the energy of line graph ℒ(G) of a graph G in terms of various graph parameters. We also obtain a relation between the signless Laplacian energy QE(G) and the incidence energy IE(G).