The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation
Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/423163 |
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| _version_ | 1832553407508905984 |
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| author | Fatih Yılmaz Durmuş Bozkurt |
| author_facet | Fatih Yılmaz Durmuş Bozkurt |
| author_sort | Fatih Yılmaz |
| collection | DOAJ |
| description | Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the
(i,j)
entry of Am (A is adjacency matrix) is equal to the number of walks of length m from vertex i to vertex j, we show that elements of mth positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers. |
| format | Article |
| id | doaj-art-7f75fe06f8374501a902621f1fe99b21 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-7f75fe06f8374501a902621f1fe99b212025-02-03T05:53:59ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/423163423163The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal RepresentationFatih Yılmaz0Durmuş Bozkurt1Department of Mathematics, Science Faculty, Selcuk University, 42250 Konya, TurkeyDepartment of Mathematics, Science Faculty, Selcuk University, 42250 Konya, TurkeyRecently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the (i,j) entry of Am (A is adjacency matrix) is equal to the number of walks of length m from vertex i to vertex j, we show that elements of mth positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers.http://dx.doi.org/10.1155/2012/423163 |
| spellingShingle | Fatih Yılmaz Durmuş Bozkurt The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation Journal of Applied Mathematics |
| title | The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation |
| title_full | The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation |
| title_fullStr | The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation |
| title_full_unstemmed | The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation |
| title_short | The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation |
| title_sort | adjacency matrix of one type of directed graph and the jacobsthal numbers and their determinantal representation |
| url | http://dx.doi.org/10.1155/2012/423163 |
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