Some new bounds for the signless Laplacian energy of a graph

TL;DR

This paper derives new bounds for the signless Laplacian energy of graphs, improving existing results and identifying extremal graphs, with additional bounds provided for regular graphs.

Contribution

It introduces improved bounds for the signless Laplacian energy and characterizes extremal graphs achieving these bounds, extending to regular graphs.

Findings

Two new lower bounds for QE(G)

One new upper bound for QE(G)

Extremal graphs characterized for the bounds

Abstract

For a simple graph $G$ with $n$ vertices, $m$ edges and signless Laplacian eigenvalues $q_{1} \geq q_{2} \geq \cdots \geq q_{n} \geq 0$, 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 $G$.