The Laplacian energy of random graphs

Gutman et al. introduced the concepts of energy E ( G ) and Laplacian energy E L ( G ) for a simple graph G, and furthermore, they proposed a conjecture that for every graph G, E ( G ) is not more than E L ( G ) . Unfortunately, the conjecture turns out to be incorrect since Liu et al. and Stevanović et al. constructed counterexamples. However, So et al. verified the conjecture for bipartite graphs. In the present paper, we obtain, for a random graph, the lower and upper bounds of the Laplacian energy, and show that the conjecture is true for almost all graphs.