University of Isfahan Transactions on Combinatorics 2251-8657 2 1 2013 03 01 New skew Laplacian energy of simple digraphs 27 37 2833 10.22108/toc.2013.2833 EN Qingqiong Cai Center for Combinatorics, nankai University, Tianjin, China Xueliang Li Center for Combinatorics, Nankai University, Tianjin 300071, China Jiangli Song Center for Combinatorics, Nankai University, Tianjin, China Journal Article 2013 04 24 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. https://toc.ui.ac.ir/article_2833_dc29c895a9ecdf11e850c68593ab72de.pdf