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Skew-spectra and skew energy of various products of graphs | ||
| Transactions on Combinatorics | ||
| مقاله 2، دوره 4، شماره 2، شهریور 2015، صفحه 13-21 اصل مقاله (238.81 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2015.6417 | ||
| نویسندگان | ||
| Xueliang Li1؛ Huishu Lian* 2 | ||
| 1Center for Combinatorics and LPMC-TJKLC, Nankai University | ||
| 2College of Science, China University of Mining and Technology | ||
| چکیده | ||
| Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. Then the spectrum of $S(G^\sigma)$ consisting of all the eigenvalues of $S(G^\sigma)$ is called the skew-spectrum of $G^\sigma$, denoted by $Sp(G^\sigma)$. The skew energy of the oriented graph $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the sum of the norms of all the eigenvalues of $S(G^\sigma)$. In this paper, we give orientations of the Kronecker product $H\otimes G$ and the strong product $H\ast G$ of $H$ and $G$ where $H$ is a bipartite graph and $G$ is an arbitrary graph. Then we determine the skew-spectra of the resultant oriented graphs. As applications, we construct new families of oriented graphs with optimum skew energy. Moreover, we consider the skew energy of the orientation of the lexicographic product $H[G]$ of a bipartite graph $H$ and a graph $G$. | ||
| کلیدواژه‌ها | ||
| skew-spectrum؛ skew energy؛ Kronecker product؛ strong product؛ lexicographic product | ||
| مراجع | ||
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C. Adiga, R. Balakrishnan and W. So (2010) The skew energy of a digraph Linear Algebra Appl. 432, 1825-1835
A. Anuradha, R. Balakrishnan, X. Chen, X. Li, H. Lian and W. So (2013) Skew spectra of oriented bipartite graphs Electron. J. Combin. 20, 12
X. Chen, X. Li and H. Lian (2013) 4-Regular oriented graphs with optimum skew energy Linear Algebra Appl. 439, 2948-2960
D. Cui and Y. Hou (2013) On the skew spectra of Cartesian products of graphs Electron. J. Combin. 20, 13
S. Gong and G. Xu (2012) 3-Regular digraphs with optimum skew energy Linear Algebra Appl. 436, 465-471
I. Gutman (1978) The energy of a graph Ber. Math.-Statist. Sekt. Forsch. Graz 103, 1-22
X. Li and H. Lian A survey on the skew energy of oriented graphs arXiv: 1304.5707
F. J. MacWilliams and N. J. A. Sloane (1977) The Theory of Error-Correcting Codes North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York-Oxford 16
G. Tian (2011) On the skew energy of orientations of hypercubes Linear Algebra Appl. 435, 2140-2149
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