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

Abstract 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 L E ( G ) , S L E ( G ) and D E ( G ) , respectively. In this paper we introduce a distance Laplacian energy D L E and distance signless Laplacian energy D S L E of a connected graph. We present Nordhaus–Gaddum type bounds on Laplacian energy L E ( G ) and signless Laplacian energy S L E ( 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 D S L E among all the graphs and trees of order n , respectively. We give lower bounds on distance Laplacian energy D L E in terms of n for graphs and trees, and characterize the extremal graphs. Also we obtain some relations between D E , D S L E and D L E of graph G . Moreover, we give several open problems in this paper.

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