NEW SKEW LAPLACIAN ENERGY OF SIMPLE DIGRAPHS

For a simple digraph G of order n with vertex set fv1; v2; : : : ; vng, let d+ i and d??i denote the out-degree and in-degree of a vertex vi in G, respectively. Let D+(G) = diag(d+ 1 ; d+ 2 ; : : : ; d+ n ) and D??(G) = diag(d??1 ; d??2 ; : : : ; d??n ). In this paper we introduce fSL(G) = eD(G)??S(G) to be a new kind of skew Laplacian matrix of G, where eD(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 fSL(G). Some lower and upper bounds of the new skew Laplacian energy are derived and the digraphs attaining these bounds are also determined.