New skew Laplacian energy of simple digraphs

Document Type: Research Paper

Authors

1 Center for Combinatorics, nankai University, Tianjin, China

2 Center for Combinatorics, Nankai University, Tianjin 300071, China

3 Center 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.

Keywords

Main Subjects


C. Adiga, R. Balakrishnan and W. So (2010). The shew energy of a digraph. Linear Algebra Appl.. 432, 1825-1835
C. Adiga and M. Smitha (2009). On the skew Laplacian energy of a digraph. Int. Math. Forum. 4 (3), 1907-1914
C. Adiga and Z. Khoshbakht (2009). On some inequalities for the skew Laplacian energgy of digraphs. JIPAM. J. Inequal. Pure Appl. Math.. 10 (3), 6
D. Cvetkovi$acute{c}$, P. Rowlinson and S. Simi$acute{c}$ (2010). An Introduction to the Theory of Graph Spectra. Cambridge Univ. Press, Cambridge.
I. Gutman (1978). The energy of a graph. Ber. Math.-Statist. Sekt. Forsch. Graz. 103, 1-22
I. Gutman and B. Zhou (2006). Laplacian energy of a graph. Linear Algebra Appl.. 414, 29-37
I. Gutman, X. Li and J. Zhang (2009). Graph Energy, in: M. Dehmer, F. Emmert-Streib (Eds.). Analysis of Complex Network: From Biology to Linguistics, Wiley-VCH Verlag, Weinheim. , 145-174
R. A. Horn and C. R. Johnson (1990). Matrix Analysis. Cambridge Univ. Press.
M. L. Kragujevac (2006). On the Laplacian energy of a graph. Czech. Math. J.. 56 (131), 1207-1213
X. Li, Y. Shi and I. Gutman (2012). Graph Energy. Springer, New York.
P. Kissani and Y. Mizoguchi (2010). Laplacian energy of directed graphs and minimizing maximum outdegree algorithms. Kyushu University Institutional Repository.