New skew Laplacian energy of simple digraphs

For a simple digraph G of order n with vertex set‎ ‎\{v_۱,v_۲,\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_۱^+,d_۲^+,\ldots,d_n^+) and‎ ‎D^-(G)=diag(d_۱^-,d_۲^-,\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.

Keywords:

energy , Laplacian energy , skew energy , skew Laplacian energy , eigenvalues