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Strong Fuzzy Chromatic Polynomial (SFCP) of Fuzzy Graphs and Some Fuzzy Graph Structures with Applications

Received: 6 December 2019     Accepted: 24 December 2019     Published: 23 January 2020
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Abstract

In fuzzy graph theory, strong arcs have separate importance. Assign different colors to the end nodes of strong arcs in the fuzzy graph is strong coloring. Strong coloring plays an important role in solving real-life problems that involve networks. In this work, we introduce the new concept, called strong fuzzy chromatic polynomial (SFCP) of a fuzzy graph based on strong coloring. The SFCP of a fuzzy graph counts the number of k-strong colorings of a fuzzy graph with k colors. The existing methods for determining the chromatic polynomial of the crisp graph are used to obtain SFCP of a fuzzy graph. We establish the necessary and sufficient condition for SFCP of a fuzzy graph to be the chromatic polynomial of its underlying crisp graph. Further, we study SFCP of some fuzzy graph structures, namely strong fuzzy graphs, complete fuzzy graphs, fuzzy cycles, and fuzzy trees. Besides, we obtain relations between SFCP and fuzzy chromatic polynomial of strong fuzzy graphs, complete fuzzy graphs, and fuzzy cycles. Finally, we present dual applications of the proposed work in the traffic flow problem. Once SFCP of a fuzzy graph is obtained, the proposed approach is simple enough and shortcut technique to solve strong coloring problems without using coloring algorithms.

Published in Pure and Applied Mathematics Journal (Volume 9, Issue 1)
DOI 10.11648/j.pamj.20200901.13
Page(s) 16-25
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Fuzzy Graph, Strong Coloring, Strong Fuzzy Chromatic Polynomial, Strong Fuzzy Graph, Complete Fuzzy Graph, Fuzzy Cycle, Fuzzy Tree, Traffic Flow Problems

References
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  • APA Style

    Mamo Abebe Ashebo, Venkata Naga Srinivasa Rao Repalle. (2020). Strong Fuzzy Chromatic Polynomial (SFCP) of Fuzzy Graphs and Some Fuzzy Graph Structures with Applications. Pure and Applied Mathematics Journal, 9(1), 16-25. https://doi.org/10.11648/j.pamj.20200901.13

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    ACS Style

    Mamo Abebe Ashebo; Venkata Naga Srinivasa Rao Repalle. Strong Fuzzy Chromatic Polynomial (SFCP) of Fuzzy Graphs and Some Fuzzy Graph Structures with Applications. Pure Appl. Math. J. 2020, 9(1), 16-25. doi: 10.11648/j.pamj.20200901.13

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    AMA Style

    Mamo Abebe Ashebo, Venkata Naga Srinivasa Rao Repalle. Strong Fuzzy Chromatic Polynomial (SFCP) of Fuzzy Graphs and Some Fuzzy Graph Structures with Applications. Pure Appl Math J. 2020;9(1):16-25. doi: 10.11648/j.pamj.20200901.13

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  • @article{10.11648/j.pamj.20200901.13,
      author = {Mamo Abebe Ashebo and Venkata Naga Srinivasa Rao Repalle},
      title = {Strong Fuzzy Chromatic Polynomial (SFCP) of Fuzzy Graphs and Some Fuzzy Graph Structures with Applications},
      journal = {Pure and Applied Mathematics Journal},
      volume = {9},
      number = {1},
      pages = {16-25},
      doi = {10.11648/j.pamj.20200901.13},
      url = {https://doi.org/10.11648/j.pamj.20200901.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200901.13},
      abstract = {In fuzzy graph theory, strong arcs have separate importance. Assign different colors to the end nodes of strong arcs in the fuzzy graph is strong coloring. Strong coloring plays an important role in solving real-life problems that involve networks. In this work, we introduce the new concept, called strong fuzzy chromatic polynomial (SFCP) of a fuzzy graph based on strong coloring. The SFCP of a fuzzy graph counts the number of k-strong colorings of a fuzzy graph with k colors. The existing methods for determining the chromatic polynomial of the crisp graph are used to obtain SFCP of a fuzzy graph. We establish the necessary and sufficient condition for SFCP of a fuzzy graph to be the chromatic polynomial of its underlying crisp graph. Further, we study SFCP of some fuzzy graph structures, namely strong fuzzy graphs, complete fuzzy graphs, fuzzy cycles, and fuzzy trees. Besides, we obtain relations between SFCP and fuzzy chromatic polynomial of strong fuzzy graphs, complete fuzzy graphs, and fuzzy cycles. Finally, we present dual applications of the proposed work in the traffic flow problem. Once SFCP of a fuzzy graph is obtained, the proposed approach is simple enough and shortcut technique to solve strong coloring problems without using coloring algorithms.},
     year = {2020}
    }
    

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    T1  - Strong Fuzzy Chromatic Polynomial (SFCP) of Fuzzy Graphs and Some Fuzzy Graph Structures with Applications
    AU  - Mamo Abebe Ashebo
    AU  - Venkata Naga Srinivasa Rao Repalle
    Y1  - 2020/01/23
    PY  - 2020
    N1  - https://doi.org/10.11648/j.pamj.20200901.13
    DO  - 10.11648/j.pamj.20200901.13
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
    SP  - 16
    EP  - 25
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20200901.13
    AB  - In fuzzy graph theory, strong arcs have separate importance. Assign different colors to the end nodes of strong arcs in the fuzzy graph is strong coloring. Strong coloring plays an important role in solving real-life problems that involve networks. In this work, we introduce the new concept, called strong fuzzy chromatic polynomial (SFCP) of a fuzzy graph based on strong coloring. The SFCP of a fuzzy graph counts the number of k-strong colorings of a fuzzy graph with k colors. The existing methods for determining the chromatic polynomial of the crisp graph are used to obtain SFCP of a fuzzy graph. We establish the necessary and sufficient condition for SFCP of a fuzzy graph to be the chromatic polynomial of its underlying crisp graph. Further, we study SFCP of some fuzzy graph structures, namely strong fuzzy graphs, complete fuzzy graphs, fuzzy cycles, and fuzzy trees. Besides, we obtain relations between SFCP and fuzzy chromatic polynomial of strong fuzzy graphs, complete fuzzy graphs, and fuzzy cycles. Finally, we present dual applications of the proposed work in the traffic flow problem. Once SFCP of a fuzzy graph is obtained, the proposed approach is simple enough and shortcut technique to solve strong coloring problems without using coloring algorithms.
    VL  - 9
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics, Wollega University, Nekemte, Ethiopia

  • Department of Mathematics, Wollega University, Nekemte, Ethiopia

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