On a conjecture of Laplacian energy of trees

Abstract

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 $G$. Radenković 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 $n$.