Cai, Q., Li, X., Song, J. (2013). New skew Laplacian energy of simple digraphs. Transactions on Combinatorics, 2(1), 27-37. doi: 10.22108/toc.2013.2833
Qingqiong Cai; Xueliang Li; Jiangli Song. "New skew Laplacian energy of simple digraphs". Transactions on Combinatorics, 2, 1, 2013, 27-37. doi: 10.22108/toc.2013.2833
Cai, Q., Li, X., Song, J. (2013). 'New skew Laplacian energy of simple digraphs', Transactions on Combinatorics, 2(1), pp. 27-37. doi: 10.22108/toc.2013.2833
Cai, Q., Li, X., Song, J. New skew Laplacian energy of simple digraphs. Transactions on Combinatorics, 2013; 2(1): 27-37. doi: 10.22108/toc.2013.2833
1Center for Combinatorics, nankai University, Tianjin, China
2Center for Combinatorics, Nankai University, Tianjin 300071, China
3Center for Combinatorics, Nankai University, Tianjin, China
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.
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