.svg){kind=logo}
DETERMINING FUZZY LABELING OF A WHEEL GRAPH AND ITS STRONG FUZZY RESOLVING SET
Keywords:
labeling, strong fuzzy resolving set, strong resolving number, wheel graphDOI:
https://doi.org/10.17654/0972087125029Abstract
Given a fuzzy set $\widetilde{V}$ on $V$, a graph $\widetilde{G}(\widetilde{V}, \widetilde{E})$ is called a fuzzy labeling graph if $\sigma: V \rightarrow[0,1]$ and $\mu: E \subseteq V \times V \rightarrow[0,1]$ are bijective membership functions such that every vertex and edge receive a unique membership degree and $\mu$ satisfies $\mu\left(u_1 u_2\right) \leq \sigma\left(u_1\right) \wedge \sigma\left(u_2\right)$ for $u_1 u_2 \in E$. A graph formed from a single vertex connected to the vertices of a cycle of length $n$ is called a wheel graph. In this research, an algorithm for fuzzy labeling of a wheel graph is constructed. We also examine the strong fuzzy resolving set (SFRS) of the fuzzy labeling wheel graph with $n+1$ vertices and get the strong resolving number $F_{s r}\left(\tilde{W}_{n+1}\right)=\left\lceil\frac{n+1}{2}\right\rceil$.
Received: May 20, 2025
Accepted: July 11, 2025
References
[1] A. N. Gani, Properties of fuzzy labeling graph, Appl. Math. Sci. 6 (2012), 3461- 3466.
[2] F. Harary and R. A. Melter, On the metric dimension of a graph, Ars. Comb. 2 (1976), 191-195.
[3] M. Jiny D and R. Shanmugapriya, Modified fuzzy resolving number, Eur. J. Mol. Clin. Med. 7(2) (2020), 5047-5052.
[4] M. Jiny D and R. Shanmugapriya, Properties of fuzzy resolving set, Turk. Comput. Math. Educ. 12(5) (2021), 1085-1089.
[5] M. Jiny D and R. Shanmugapriya, The connectedness of fuzzy graph and the resolving number of fuzzy digraphs, Fuzzy Intelligent Systems: Methodologies, Techniques and Applications, Wiley Publication, 2021.
[6] M. Jiny D and R. Shanmugapriya, Fuzzy resolving number and real basis generating fuzzy graphs, Afrika Mat. 34(25) (2023), 1-13.
[7] S. Mathew, J. N. Mordeson and D. S. Malik, Fuzzy Graph Theory, Springer International Publishing, Switzerland AG, 2018.
[8] C. O. F. Parera, M. Salmin, A. W. Bustan and R. Mahmud, Application of metric dimensions to minimize the installation of fire sensors on the Rectorate building of Pasifik Morotal University, Proc. ICST-2022, MATEC Web Conf. 372 (2022), Article No. 04005, pp. 1-3.
[9] S. Prabhu, V. Manimozhi, M. Arulperumjothi and S. Klavžar, Twin vertices in fault-tolerant metric sets and fault-tolerant metric dimension of multistage interconnection networks, Appl. Math. Comput. 420 (2022), 126897.
[10] A. Rosenfeld, Fuzzy graphs, Fuzzy Sets and their Applications to Cognitive and Decision Processes, L. A. Zadeh, K. S. Fu and M. Shimura, eds., Academic Press. 1975, pp. 77-95.
[11] I. Rosyida, W. F. Albar, M. Habiburrohman and M. Jiny D, On the construction of a strong fuzzy resolving set of fuzzy graphs, Power Syst. Technol. 48(2) (2024), 495-504.
[12] P. J. Slater, Leaves of trees, Congr. Numer. 14 (1975), 549-559.
[13] R. Shanmugapriya and M. Jiny D, Fuzzy super resolving number of fuzzy labelling graphs, J. Adv. Res. Dyn. Control Syst. 7 (2019), 606-611.
[14] R. Shanmugapriya and M. Jiny D, Fuzzy super resolving number and resolving number of some special graphs, TWMS J. App. and Eng. Math. 11(2) (2021), 459-468.
[15] R. Shanmugapriya and M. Jiny D, On regular fuzzy resolving set, TWMS J. App. and Eng. Math. 12(4) (2022), 1322-1328.
[16] R. Shanmugapriya, M. Vasuki and P. K. Hemalatha, Average resolving number of a fuzzy graph and its application in network theory, Recent Trends in Computational Intelligence and its Application, D. Sugumaran, P. Souvik, L. Dac-Nhuong and N. Z. Jhanjhi, eds., CRC Press, 2023, pp. 1-5.
[17] R. Shanmugapriya, P. K. Hemalatha, L. Cepova and J. Struz, A study of independency on fuzzy resolving sets of labelling graphs, Mathematics 11(16) (2023), 3440.
[18] R. Shanmugapriya and M. Vasuki, A novel case on fuzzy resolving sets on interval-valued fuzzy graph and its application on signal processing unit, Adv. Nonlinear Var. Inequal. 23(3s) (2025), 547-557.
[19] R. C. Tillquist, R. M. Frongillo and M. E. Lladser, Getting the lay of the land in discrete space: A survey of metric dimension and its applications, SIAM Rev. 65(4) (2023), 919-962.
[20] M. Vasuki, R. Shanmugapriya, M. Mahdal and R. Cep, A study on fuzzy resolving domination sets and their application in network theory, Mathematics 11(2) (2023), 317.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Copyright ©2025 The Author(s)
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________
Attribution: Credit Pushpa Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
Contact Puspha Publishing House for more info or permissions.
