Group for Research in Decision Analysis

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

## Kinkar Chandra Das, Mustapha Aouchiche, and Pierre Hansen

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