New skew Laplacian energy of a simple digraph

Abstract

For a simple digraph $G$ of order $n$ with vertex set $\{v_1,v_2,\dots ,v_n\}$, let $d_i^{+}$ and $d_i^{-}$ denote the out-degree and in-degree of a vertex $v_i$ in $G$, respectively. Let $D^{+}(G)=diag(d_1^{+},d_2^{+},\dots ,d_n^{+})$ and $D^{-}(G)=diag(d_1^{-},d_2^{-},\dots ,d_n^{-})$. In this paper we introduce $\widetilde{SL}(G)=\tilde{D}(G)-S(G)$ 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.

13 pages