Group for Research in Decision Analysis

G-2017-96

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

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Let \(G\) be a graph of order \(n\). The energy \(\mathcal{E}(G)\) 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.

, 21 pages