On (distance) Laplacian energy and (distance) signless Laplacian energy of graphs

Let G be a graph of order n. The energy E(G) of a simple graph G is the sum of absolute values of the eigenvalues of its adjacency matrix. The Laplacian energy, the signless Laplacian energy and the distance energy of graph G are denoted by LE(G), SLE(G) and DE(G), respectively. In this paper we introduce a distance Laplacian energy DLE and distance signless Laplacian energy DSLE of a connected graph. We present Nordhaus–Gaddum type bounds on Laplacian energy LE(G) and signless Laplacian energy SLE(G) in terms of order n of graph G and characterize graphs for which these bounds are best possible. The complete graph and the star give the smallest distance signless Laplacian energy DSLE among all the graphs and trees of order n, respectively. We give lower bounds on distance Laplacian energy DLE in terms of n for graphs and trees, and characterize the extremal graphs. Also we obtain some relations between DE, DSLE and DLE of graph G. Moreover, we give several open problems in this paper.