Lattice valued fuzzy graphs

Abstract

Fuzzy graphs have numerous applications in real life. Up till now, there have been many extensions of fuzzy graphs. For example,  intuitionistic fuzzy graphs,  Pythagorean fuzzy graphs, Fermatean fuzzy graphs,  $q$-rung orthopair fuzzy graphs, picture fuzzy graphs, spherical fuzzy graphs, $q$-rung picture fuzzy graphs, interval valued fuzzy graphs, single valued neutrosophic fuzzy graphs, bipolar fuzzy graphs, $m$-polar fuzzy graphs, and so on. In the present paper, we shall present a novel definition of lattice-valued fuzzy graphs by lattice  implication operators $\mapsto$,  and  prove that  various fuzzy graphs mentioned above can be regarded as  special lattice-valued fuzzy graphs. Moreover we shall present some characterizations of lattice-valued fuzzy graphs.

Keywords

• lattice valued fuzzy set
• lattice valued fuzzy graph
• L-fuzzy relation
• q-rung orthopair fuzzy graphs
• q-rung picture fuzzy graphs.

References

1. [1] M. Akram, Bipolar fuzzy graphs, Inf. Sci. 181, 5548–5564, 2011.
2. [2] M. Akram, R. Akmal and N. Alshehri, On m-polar fuzzy graph structures, Springer- Plus. 5, 1448, 2016.
3. [3] M. Akram and B. Davvaz, Strong intuitionistic fuzzy graphs, Filomat, 26, 177–196, 2012.
4. [4] M. Akram and W.A. Dudek, Interval-valued fuzzy graphs, Comput Math Appl 61, 289–299, 2011.
5. [5] M. Akram and A. Habib, q-rung picture fuzzy graphs: a creative view on regularity with applications, J Appl Math Comput. 61, 235–280, 2019.
6. [6] M. Akram and S. Naz, Energy of Pythagorean fuzzy graphs with applications, Mathematics. 6, 136, 2018. https://doi.org/10.1007/s12190-019-01249-y
7. [7] M. Akram, D. Saleem and T. A. Hawary, Spherical fuzzy graphs with application to decision-making, Math. Comput. Appl. 25, 8, 2020.
8. [8] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20, 87–96, 1986.

9. [9] S. Broumi, F. Smarandache, M. Talea and A. Bakali, Single valued neutrosophic graphs: Degree, order and size, 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE). Vancouver, BC, Canada, 2444–2451, 2016.
10. [10] J. Chen, S. Li, S. Ma and X. Wang, m-polar fuzzy sets: an extension of bipolar fuzzy sets, Sci World J. 2014, 8, 2014.
11. [11] B.C. Cuong, Picture fuzzy sets, Journal of Computer Science and Cybernetics, 30, 409–420, 2014.
12. [12] B.C. Cuong and V. Kreinovich, Picture fuzzy sets–A new concept for computational intelligence problems, 2013 ThirdWorld Congress on Information and Communication Technologies (WICT 2013). Hanoi, Vietnam, 1-6, 2013.
13. [13] G. Ghorai and M. Pal, On some operations and density of m-polar fuzzy graphs, Pacific Science Review A: Natural Science and Engineering. 17, 14–22, 2015.
14. [14] G. Ghorai and M. Pal, A study on m-polar fuzzy planar graphs, Int J Comput Sci Math. 7, 283–292, 2016.
15. [15] G. Ghorai and M. Pal, Faces and dual of m-polar fuzzy planar graphs, J Intell Fuzzy Syst. 31, 2043–2049, 2016.
16. [16] P. Giri, S. Amanathulla and K. M. Das, An analysis of Fermatean fuzzy graph and its application in a car company, J Appl Math Comput. 70, 3575–3602, 2024.
17. [17] J.A. Goguen, L-fuzzy sets, J Math Anal Appl. 18, 145–174, 1967.
18. [18] A. Habib, M. Akram and A. Farooq, q-Rung Orthopair Fuzzy Competition Graphs with Application in the Soil Ecosystem, Mathematics. 7, 91, 2019.
19. [19] M.K. Hasan, A. Sultana and N.K. Mitra, Picture fuzzy relations over picture fuzzy sets, American Journal of Computational Mathematics, 13, 161–184, 2023.
20. [20] H.-L. Huang and F.-G. Shi, L-fuzzy numbers and their properties, Inf. Sci., 178, 1141–1151, 2008.
21. [21] U. Jana and G. Ghorai, First Entire Zagreb Index of fuzzy graph and its application, Axioms. 12, 415, 2023.
22. [22] U. Jana and G. Ghorai, Gutman index of fuzzy graphs with application, Soc. Netw. Anal. Min. 14, 209, 2024.
23. [23] U. Jana and G. Ghorai, On the neighborhood inverse sum indeg index of fuzzy graph with application, J. Appl. Math. Comput. 70, 1211–1239, 2024.
24. [24] G.F. Kutlu and C. Kahraman, Spherical fuzzy sets and spherical fuzzy TOPSIS method, J Intell Fuzzy Syst. 36, 337–352, 2019.
25. [25] L. Li, R. Zhang, J. Wang, X. Shang and K. Bai, A novel approach to muti-attribut group decision making with q-rung picture linguistic information, Symmetry. 10, 172, 2018.
26. [26] S. Naz, S. Ashraf and M. Akram, A novel approach to decision making with Pythagorean fuzzy information, Mathematics. 6, 95, 2018.
27. [27] M. Pal, S. Samanta and G. Ghorai, Modern Trends in Fuzzy Graph Theory, Springer Singapore, 2020.
28. [28] R. Parvathi and M.G. Karunambigai, Intuitionistic Fuzzy Graphs, In: Reusch, B. (eds) Computational Intelligence, Theory and Applications, Springer, Berlin, Heidelberg. 38, 2006.
29. [29] R. Parvathi, M.G. Karunambigai and K.T. Atanassov, Operations on intuitionistic fuzzy graphs, IEEE International Conference on Fuzzy Systems, Jeju, Korea. 1396–1401, 2009.
30. [30] S. Rebecca, G.J. Kumar, S.K. Lydia and J. Malathi, An introduction to Fermatean fuzzy graph structure and its applications in decision making, Nanotechnology Perceptions. 20, 3235–3251, 2024.
31. [31] A. Rosenfeld, Fuzzy Graphs, in: Fuzzy Sets and Their Applications to Cognitive and Decision Processes. Proceedings of the US–Japan Seminar on Fuzzy Sets and their Applications. Elsevier, 77–95, 1975.
32. [32] M.K. Roy and R. Biswas, I-v fuzzy relations and Sanchez’s approach for medical diagnosis, Fuzzy Sets Syst. 47, 35–38, 1992.
33. [33] T. Senapati and R.R. Yager, Fermatean fuzzy sets, Journal of Ambient Intelligence and Humanized Computing. 11, 663–674, 2020.
34. [34] F.-G. Shi, Theory of $L_\alpha$-nest sets and $L_\beta$-nest sets and their applications, Fuzzy Syst Math. 4, 65–72, 1995. (in Chinese).
35. [35] F.-G. Shi, L-fuzzy relation and L-fuzzy subgroup, J. Fuzzy Math. 8, 491–499, 2000.
36. [36] F.-G. Shi, (L,M)-fuzzy matroids, Fuzzy Sets Syst. 160, 2387–2400, 2009.
37. [37] F.-G. Shi, A new approach to the fuzzification of matroids, Fuzzy Sets Syst. 160, 696–705, 2009.
38. [38] F.-G. Shi and Z.-Y. Xiu, (L,M)-fuzzy convex structures, J. Nonlinear Sci. Appl. 10, 3655–3669, 2017.
39. [39] F. Smarandache, Neutrosophy: neutrosophic probability, set, and logic: analytic synthesis & synthetic analysis, Rehoboth: American Research Press, 1998.
40. [40] F. Smarandache, A unifying field in logics. Neutrosophy: Neutrosophic probability, set and logic, Rehoboth: American Research Press, 1999.
41. [41] F. Smarandache, Neutrosophic set–a generalization of the intuitionistic fuzzy set, 2006 IEEE International Conference on Granular Computing. Atlanta, GA, USA, 38–42, 2006.
42. [42] R. Verma, J.M. Merigo and M. Sahni, Pythagorean fuzzy graphs: Some results, arXiv. 2018.
43. [43] G.J. Wang, Theory of topological molecular lattices, Fuzzy Sets Syst. 47, 351–376, 1992.
44. [44] H. Wang, F. Smarandache, Y.Q. Zhang and R. Sunderraman, Single valued neutrosophic sets, Multisp. Multistruct. 4, 410–413, 2010.
45. [45] Y.-F. Wen, Y. Zhong, F.-G. Shi, L-fuzzy convexity induced by L-convex degree on vector spaces, J Intell Fuzzy Syst. 33, 4031–4041, 2017.
46. [46] R.R. Yager, Pythagorean fuzzy subsets, 2013 Joint IFSAWorld Congress and NAFIPS Annual Meeting (IFSA/NAFIPS). Edmonton, AB, Canada, 57–61, 2013. Int J Intell Syst. 28, 436–452, 2013.
47. [47] R.R. Yager, Generalized orthopair fuzzy sets, IEEE Trans Fuzzy Syst. 25, 1222–1230, 2017.
48. [48] R.R. Yager and A.M. Abbasov, Pythagorean membership grades, complex numbers, and decision making,
49. [49] L.A. Zadeh, Fuzzy sets, Information and Control. 8, 238–353, 1965.
50. [50] L.A. Zadeh, The concept of a linguistic and application to approximate reasoning-I, Inf. Sci. 3, 199–249, 1975.
51. [51] W.-R. Zhang, Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis, 1994 Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference. San Antonio, TX, USA, 305–309, 1994.
52. [52] W.-R. Zhang, Bipolar fuzzy sets, 1998 IEEE International Conference on Fuzzy Systems Proceedings. Anchorage, AK, USA. 835–840, 1998.
53. [53] C. Zuo, A. Pal and A. Dey, New concepts of picture fuzzy graphs with application, Mathematics. 7, 470, 2019.