Some new bounds for the signless Laplacian energy of a graph

Abstract

For a simple graph $G$ with $n$ vertices, $m$ edges and signless Laplacian eigenvalues $q_1\ge q_2\ge \cdots \ge q_n\ge 0$, its the signless Laplacian energy $QE(G)$ is defined as $QE(G)={\sum }_{i=1}^n|q_i-\bar{d}|$, where $\bar{d}=\frac{2m}{n}$ is the average vertex degree of $G$. In this paper, we obtain two lower bounds ( see Theorem 3.1 and Theorem 3.2 ) and one upper bound for $QE(G)$ ( see Theorem 3.3 ), which improve some known bounds of $QE(G)$, and moreover, we determine the corresponding extremal graphs that achieve our bounds. By subproduct, we also get some bounds for $QE(G)$ of regular graph $G$.

19 pages