Relation between signless Laplacian energy, energy of graph and its line graph

Abstract

The energy of a simple graph G, E(G), is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of line graph and the signless Laplacian energy of graph G are denoted by E(LG) (LG is the line graph of G) and LE+(G), respectively. In this paper we obtain a relation between E(LG) and LE+(G) of graph G. From this relation we characterize all the graphs satisfying E(LG)=LE+(G)+4mn−4. We also present a relation between E(G) and E(LG). Moreover, we give an upper bound on E(LG) of graph G and characterize the extremal graphs.