more on the bounds for the skew laplacian energy of weighted digraphs

Let $mathscr{d}$ be a simple connected digraph with $n$ vertices and $m$ arcs and let $w(mathscr{d})=mathscr{d},w)$ be the weighted digraph corresponding to $mathscr{d}$, where the weights are taken from the set of non-zero real numbers. let $nu_1,nu_2, dots,nu_n$ be the eigenvalues of the skew laplacian weighted matrix $widetilde{sl}w(mathscr{d})$ of the weighted digraph $w(mathscr{d})$. in this paper, we discuss the skew laplacian energy $widetilde{sle}w(mathscr{d})$ of weighted digraphs and obtain the skew laplacian energy of the weighted star $w(mathscr{k}_{1, n})$ for some fixed orientation to the weighted arcs. we obtain lower and upper bounds for $widetilde{sle}w(mathscr{d})$ and show the existence of weighted digraphs attaining these bounds.