More on the bounds for the skew Laplacian energy of weighted digraphs

Abstract

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.