New skew Laplacian energy of simple digraphs

For a simple digraph $G$ of order $n$ with vertex set${v_1,v_2,ldots, v_n}$, let $d_i^+$ and $d_i^-$ denote theout-degree and in-degree of a vertex $v_i$ in $G$, respectively. Let$D^+(G)=diag(d_1^+,d_2^+,ldots,d_n^+)$ and$D^-(G)=diag(d_1^-,d_2^-,ldots,d_n^-)$. In this paper we introduce$widetilde{SL}(G)=widetilde{D}(G)-S(G)$ to be a new kind of skewLaplacian matrix of $G$, where $widetilde{D}(G)=D^+(G)-D^-(G)$ and$S(G)$ is the skew-adjacency matrix of $G$, and from which we definethe skew Laplacian energy $SLE(G)$ of $G$ as the sum of the norms ofall the eigenvalues of $widetilde{SL}(G)$. Some lower and upperbounds of the new skew Laplacian energy are derived and the digraphsattaining these bounds are also determined.