TY - JOUR

T1 - On (distance) Laplacian energy and (distance) signless Laplacian energy of graphs

AU - Das, Kinkar Ch

AU - Aouchiche, Mustapha

AU - Hansen, Pierre

N1 - Funding Information:
The authors are much grateful to two anonymous referees for their valuable comments on our paper, which have considerably improved the presentation of this paper. The first author was supported by GERAD and the Data Mining Chair of HEC Montréal for hospitality. Moreover, the first author is supported by the Sungkyun research fund, Sungkyunkwan University , 2017, and National Research Foundation of the Korean government with Grant No. 2017R1D1A1B03028642 .
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2018/7/10

Y1 - 2018/7/10

N2 - 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.

AB - 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.

KW - (Signless) Laplacian energy

KW - Distance (signless) Laplacian eigenvalues

KW - Distance (signless) Laplacian energy

KW - Distance eigenvalues

KW - Distance energy

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U2 - 10.1016/j.dam.2018.01.004

DO - 10.1016/j.dam.2018.01.004

M3 - Article

AN - SCOPUS:85044499110

VL - 243

SP - 172

EP - 185

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -