On a conjecture of Laplacian energy of trees

Let $G$ be a simple graph with $n$ vertices, $m$ edges having Laplacian eigenvalues $\mu_1, \mu_2, \dots, \mu_{n-1},\mu_n=0$. The Laplacian energy $LE(G)$ is defined as $LE(G)=\sum_{i=1}^{n}|\mu_i-\overline{d}|$, where $\overline{d}=\frac{2m}{n}$ is the average degree of $G$. Radenkovi\'{c} and Gutman conjectured that among all trees of order $n$, the path graph $P_n$ has the smallest Laplacian energy. Let $\mathcal{T}_{n}(d)$ be the family of trees of order $n$ having diameter $d$. In this paper, we show that Laplacian energy of any tree $T\in \mathcal{T}_{n}(4)$ is greater than the Laplacian energy of $P_n$, thereby proving the conjecture for all trees of diameter $4$. We also show the truth of conjecture for all trees with number of non-pendent vertices at most $\frac{9n}{25}-2$. Further, we give some sufficient conditions for the conjecture to hold for a tree of order $n$.