Skew-spectra and skew energy of various products of graphs
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 e...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Isfahan
2015-06-01
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| Series: | Transactions on Combinatorics |
| Subjects: | |
| Online Access: | http://www.combinatorics.ir/pdf_6417_49554edbea5b1672fe3293c44812de6d.html |
| _version_ | 1811204528789258240 |
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| author | Xueliang Li Huishu Lian |
| author_facet | Xueliang Li Huishu Lian |
| author_sort | Xueliang Li |
| collection | DOAJ |
| description | 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 $Hotimes G$ and the strong
product $Hast G$ 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 $G$. |
| first_indexed | 2024-04-12T03:14:44Z |
| format | Article |
| id | doaj.art-de6127314a734405996df4edc5926b33 |
| institution | Directory Open Access Journal |
| issn | 2251-8657 2251-8665 |
| language | English |
| last_indexed | 2024-04-12T03:14:44Z |
| publishDate | 2015-06-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | Transactions on Combinatorics |
| spelling | doaj.art-de6127314a734405996df4edc5926b332022-12-22T03:50:14ZengUniversity of IsfahanTransactions on Combinatorics2251-86572251-86652015-06-01421321Skew-spectra and skew energy of various products of graphsXueliang Li0Huishu Lian 1Center for Combinatorics and LPMC-TJKLC, Nankai UniversityCollege of Science, China University of Mining and TechnologyGiven 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 $Hotimes G$ and the strong product $Hast G$ 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 $G$.http://www.combinatorics.ir/pdf_6417_49554edbea5b1672fe3293c44812de6d.htmlskew-spectrumskew energyKronecker productstrong productlexicographic product |
| spellingShingle | Xueliang Li Huishu Lian Skew-spectra and skew energy of various products of graphs Transactions on Combinatorics skew-spectrum skew energy Kronecker product strong product lexicographic product |
| title | Skew-spectra and skew energy of various products of graphs |
| title_full | Skew-spectra and skew energy of various products of graphs |
| title_fullStr | Skew-spectra and skew energy of various products of graphs |
| title_full_unstemmed | Skew-spectra and skew energy of various products of graphs |
| title_short | Skew-spectra and skew energy of various products of graphs |
| title_sort | skew spectra and skew energy of various products of graphs |
| topic | skew-spectrum skew energy Kronecker product strong product lexicographic product |
| url | http://www.combinatorics.ir/pdf_6417_49554edbea5b1672fe3293c44812de6d.html |
| work_keys_str_mv | AT xueliangli skewspectraandskewenergyofvariousproductsofgraphs AT huishulian skewspectraandskewenergyofvariousproductsofgraphs |