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 |
Fuzzy Graph, Strong Coloring, Strong Fuzzy Chromatic Polynomial, Strong Fuzzy Graph, Complete Fuzzy Graph, Fuzzy Cycle, Fuzzy Tree, Traffic Flow Problems
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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
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
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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Download@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}
}
TY - JOUR 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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