New skew Laplacian energy of a simple digraph

For a simple digraph $G$ of order $n$ with vertex set ${v_1,v_2,\ldots, v_n}$,let$d_i^+$and$d_i^-$denotetheout-degreeandin-degreeofavertex$v_i$in$G$,respectively\mathrm{.}Let$D^+(G)=diag(d_1^+,d_2^+,\ldots,d_n^+)$and$D^-(G)=diag(d_1^-,d_2^-,\ldots,d_n^-)$\mathrm{.}Inthispaperweintroduce$\widetilde{SL}(G)=\widetilde{D}(G)-S(G)$ to be a new kind of skew Laplacian matrix of $G$, where $\tilde{D}(G)=D^{+}(G)-D^{-}(G)$ and $S(G)$ is the skew-adjacency matrix of $G$, and from which we define the skew Laplacian energy $SLE(G)$ of $G$ as the sum of the norms of all the eigenvalues of $\widetilde{SL}(G)$. Some lower and upper bounds of the new skew Laplacian energy are derived and the digraphs attaining these bounds are also determined.