On Signless Laplacian Energy and its Variations

Abstract

The signless Laplacian energy of a simple connected graph $G$ of order n and size m is defined in [1] as $LE^{+}(G) = \sum\limits_{i=1}^{n} \mid {\lambda_{i} - \frac{2m}{n}} \mid $, where $\lambda_{i}, i=1,2,\dots ,n$ are the eigen values of the signless Laplacian matrix $L^{+}(G)$. In this paper, a new concept called signless Laplacian maximum eccentricity energy is introduced and value of the same is determined for some families of graphs. Also, signless Laplacian energy and signless Laplacian maximum eccentricity energy are compared for some classes of graphs. Further, signless Laplacian energy of direct product of graphs is obtained.