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Volume 9, Issue 1, February 2020, Page: 16-25
Strong Fuzzy Chromatic Polynomial (SFCP) of Fuzzy Graphs and Some Fuzzy Graph Structures with Applications
Mamo Abebe Ashebo, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Venkata Naga Srinivasa Rao Repalle, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Received: Dec. 6, 2019;       Accepted: Dec. 24, 2019;       Published: Jan. 23, 2020
DOI: 10.11648/j.pamj.20200901.13      View  38      Downloads  44
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.
Keywords
Fuzzy Graph, Strong Coloring, Strong Fuzzy Chromatic Polynomial, Strong Fuzzy Graph, Complete Fuzzy Graph, Fuzzy Cycle, Fuzzy Tree, Traffic Flow Problems
To cite this article
Mamo Abebe Ashebo, Venkata Naga Srinivasa Rao Repalle, Strong Fuzzy Chromatic Polynomial (SFCP) of Fuzzy Graphs and Some Fuzzy Graph Structures with Applications, Pure and Applied Mathematics Journal. Vol. 9, No. 1, 2020, pp. 16-25. doi: 10.11648/j.pamj.20200901.13
Copyright
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
Zadeh, L. A, Fuzzy sets. Inf. Control, vol. 8, No. 3, 1965, pp. 338-353.
[2]
Rosenfeld, A, Fuzzy graphs, Fuzzy Sets, and Their Applications; Zadeh, L. A, Fu, K. S., Shimura, M., Eds.; Academic Press: New York, NY, USA, 1975; pp. 77-95.
[3]
Bhutani, K, R., On automorphisms of fuzzy Graphs. Pattern Recognit. Lett. vol. 9, No. 3, 1989, pp. 159-162.
[4]
Mordeson, J. N., and Nair, P. S. Cycles and cocycles of fuzzy graphs. Inf. Sci. vol. 90, 1996, pp. 39-49.
[5]
Sunitha, M. S., and Vijayakumar, A. A., Characterization of fuzzy trees. Inf. Sci. vol. 113, 1999, pp. 293-300.
[6]
Bhutani, K. R. and Rosenfeld, A., Strong arcs in fuzzy graphs. Inf. Sci. vol. 152, 2003, pp. 319- 322.
[7]
Mathew, S. and Sunitha, M. S. Types of arcsin a fuzzy graph. Inf. Sci. vol. 179, No. 11, 2009, pp. 1760-1768.
[8]
Mathew, S. and Sunitha, M. S. Strongest strong cycles and theta fuzzy graphs. IEEE Trans. Fuzzy Syst. vol. 21, No. 6, 2013, pp. 1096-1104.
[9]
Mordeson, J. N.; Mathew, S. Advanced topics in fuzzy graph theory. Springer Nature: Gewerbestrasse 11, 6330 Cham, Switzerland, pp. 15-54, 2019.
[10]
Talebi, A. A. Cayley fuzzy graphs on the fuzzy groups. Comput. Appl. Math. vol. 37, No. 4, 2018, pp. 4611-4632.
[11]
Dhanyamol, M. V. On certain transit functions in fuzzy graphs. Int. J. Uncertainty Fuzziness Knowl. -Based Syst. Vol. 25, No. 6, 2017, pp. 917-928.
[12]
Tom, M. and Sunitha, M. S. Strong sum distance in fuzzy graphs. SpringerPlus vol. 4, No. 214, 2015, pp. 1-14.
[13]
Mathew, S., Yang, H. L., and Mathew, J. K. Saturation in fuzzy graphs. New Math. Nat. Comput. Vol. 14, No. 1, 2018, pp. 113-128.
[14]
Harinath, P. and Lavanya, S., Fuzzy graph Structures. Int. J. Appl. Eng. Res. vol. 10, 2015, pp. 70–74.
[15]
Sitara, M., Akram, M., and Bhatti, M. Y., Fuzzy graph structures with application, Mathematics, vol. 7, No. 1, 2019, pp. 63.
[16]
Akram, M. and Sitara, M., Certain fuzzy graph structures, J. Appl. Math. Comput., vol. 61, 2019, pp. 25-56.
[17]
Akram, M., Bipolar fuzzy graphs, Inf. Sci. vol. 181, No. 24, pp. 5548-5564, 2011.
[18]
Akram, M. m-polar fuzzy graphs: Theory, Methods & Applications. Springer Nature: Gewerbestrasse 11, 633, Cham, Switzerland, pp. 7-112, 2019
[19]
Verma, R., Merigo, J. M. and Sahni, M., Pythagorean fuzzy graphs: some results, arXiv: 1806. 06721v1.
[20]
Ashraf, S., Naz, S., and Kerre, E. E., Dombi fuzzy graphs. Fuzzy Inf. Eng. vol. 10, 2018, pp. 58–79.
[21]
Akram, M., Dar, J. M., and Naz, S., Pythagorean Dombi fuzzy graphs. Complex Intell. Syst. 2019, pp. 1-26.
[22]
Zuo, C., Pal, A., and Dey, A., New concepts of picture fuzzy graphs with application. Mathematics, vol. 7, No. 5, 2019, pp. 470,
[23]
Akram, M. and Waseem, N., Novel applications of bipolar fuzzy graphs to decision-making problems. J. Appl. Math. Comput. vol. 56, 2018, pp. 73-91.
[24]
Naz, S., Ashraf, S., and Akram, M., A novel approach to decision making with Pythagorean fuzzy information, Mathematics, vol. 6, No. 6, 2018, pp. 95.
[25]
Akram, M. and Habib, A., Specific types of Pythagorean fuzzy graphs and application to decision making. Math. Comput. Appl., vol. 23, No. 3, 2018, pp. 42.
[26]
Mordeson, J. N.; Mathew, S.; Malik, D. S. Fuzzy graph theory with applications to human trafficking. Springer International Publishing: Gewerbestrasse 11, 6330 Cham, Switzerland, pp. 181-107, 2018.
[27]
Binu, M., Mathew, S. and Mordeson, J. N., Connectivity index of a fuzzy graph and its application to human trafficking. Fuzzy sets Syst. vol. 360, 2019, pp. 117-136.
[28]
Binu, M., Mathew, S., and Mordeson, J. N., Wiener index of a fuzzy graph and application to illegal immigration networks. Fuzzy Sets Syst. In Press. 2019.
[29]
Muñoz, S., Ortuño, M. T., Ramírez, J. and Yáñez, J., Coloring fuzzy graphs. Omega, vol. 33, No. 3, 2005, pp. 211–221.
[30]
Eslahchi, C. and Onagh, B. N., Vertex strength of fuzzy graphs. Int. J. Math. Math. Sci. vol. 2006 (Article ID 43614): 9, 2006.
[31]
Kishore, A. and Sunitha, M. S., Chromatic number of fuzzy graphs. Ann. Fuzzy Math. Inform. vol. 7, No. 4, 2014, pp. 543-551,
[32]
Samanta, S., Pramanik, T. and Pal, M., Fuzzy coloring of fuzzy graphs Afr. Mat. vol. 27, 2016, 2016, pp. 37-50.
[33]
Mahapatra, T. and Pal, M., Fuzzy colouring of m-polar fuzzy graph and its application. J. Intell. Fuzzy Syst. vol. 35, 2018, pp. 6379-6391.
[34]
Kishore, A. and Sunitha, M. S., Strong chromatic number of fuzzy graphs, Ann. Pure Appl. Math. vol. 7, 2014, pp. 52-60.
[35]
Rosyida, I., Widodo, Indrani, Ch. R., Indriati, D., and Nurhaida. Fuzzy chromatic number of union of fuzzy graphs: An algorithm, properties, and its application Fuzzy sets Syst. in Press, 2019.
[36]
Mamo, A. A. and Srinivasa Rao, R. V. N., Fuzzy chromatic polynomial of fuzzy graphs with crisp and fuzzy vertices using α-cuts. Advances in Fuzzy Systems, vol. 2019, Article ID 5213020, 11 pages.
[37]
Mordeson, J. N. and Nair, P. S., Fuzzy Graphs and Fuzzy Hypergraphs; Springer: Heidelberg, Germany, pp. 19-39, 2000.
[38]
Mathew, S., Mordeson, J. N. and Malik, D. S., Fuzzy graph Theory, Springer International Publishing: Gewerbestrasse 11, 6330 Cham, Switzerland, pp. 13-57, 2018.
[39]
Nagoorgani, A., Isomorphism properties on strong fuzzy graphs. Int. J. Algorithm Comput. Math. vol. 2, No. 1, pp. 39-47, 2009.
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