A note on some lower bounds of the Laplacian energy of a graph

Abstract

‎‎‎For a simple connected graph $G$ of order $n$ and size $m$‎, ‎the Laplacian energy of $G$ is defined as‎ ‎$LE(G)=\sum_{i=1}^n|\mu_i-\frac{2m}{n}|$ where $\mu_1‎, ‎\mu_2,\ldots‎,‎‎\mu_{n-1}‎, ‎\mu_{n}$‎ ‎are the Laplacian eigenvalues of $G$ satisfying $\mu_1\ge \mu_2\ge\cdots \ge \mu_{n-1}>‎ ‎\mu_{n}=0$‎. ‎In this note‎, ‎some new lower bounds on the graph invariant $LE(G)$ are derived‎. ‎The obtained results are compared with some already known lower bounds of $LE(G)$‎.

Keywords

• ‎laplacian eigenvalue‎
• ‎laplacian energy (of a‎ ‎graph)‎
• ‎first zagreb index