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`https://scholar.ptuk.edu.ps/handle/123456789/202`

Title: | Minimum Flows in the Total Graph of a Commutative Ring |

Authors: | Sander, Torsten Nazzal, Khalida |

Keywords: | Constant-sum k-flow; minimum flow; the ring of integers modulo n; total graph of a commutative ring; zero-sum k-flow Main Subjects |

Issue Date: | 30-Jan-2014 |

Publisher: | Transactions on Combinatorics |

Citation: | , and Khalida Nazzal . Transactions on Combinatorics Vol. (2014). 3 No. 3, 11-20 |

Series/Report no.: | 3 No. 3;11-20 |

Abstract: | Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T(Γ(R)), is the simple (undirected) graph with vertex set R where two distinct vertices are adjacent if their sum lies in Z(R). This work considers minimum zero-sum k-flows for T(Γ(R)). Both for |R| even and the case when |R| is odd and Z(G) is an ideal of R it is shown that T(Γ(R)) has a zero-sum 3-flow, but no zero-sum 2-flow. As a step towards resolving the remaining case, the total graph T(Γ(Zn)) for the ring of integers modulo n is considered. Here, minimum zero-sum k-flows are obtained for n=pr and n=prqs (where p and q are primes, r and s are positive integers). Minimum zero-sum k-flows as well as minimum constant-sum k-flows in regular graphs are also investigated. |

Description: | Article |

URI: | https://scholar.ptuk.edu.ps/handle/123456789/202 |

Appears in Collections: | Applied science faculty |

Files in This Item:

File | Description | Size | Format | |
---|---|---|---|---|

560254153d38ef1a1f5b472f831c835f.pdf | Main article | 4.97 MB | Adobe PDF | View/Open |

TOC_2014 Autumn_Vol 3_Issue 3_Pages 11-20.pdf | 234.45 kB | Adobe PDF | View/Open |

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