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

‎‎‎For a simple connected graph $G$ of order $n$ and size $m$‎, ‎the Laplacian energy of $G$ is defined as‎ ‎$LE(G)=\sum_{i=۱}^n|\mu_i-\frac{۲m}{n}|$ where $\mu_۱‎, ‎\mu_۲,\ldots‎,‎‎\mu_{n-۱}‎, ‎\mu_{n}$‎ ‎are the Laplacian eigenvalues of $G$ satisfying $\mu_۱\ge \mu_۲\ge\cdots \ge \mu_{n-۱}>‎ ‎\mu_{n}=۰$‎. ‎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