New skew Laplacian energy of simple digraphs

Abstract

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 the‎ ‎out-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 skew‎ ‎Laplacian 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 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.