Energy and Laplacian Energy of Graphs under the Bipolar Valued Hesitant Fuzzy Framework

Sakshi Dev Pandey ¹ ( )
Abhay S. Ranadive ² ( )
Sovan Samanta ³ ( )
Leo Mrsic ⁴ ( )
Antonios Kalampakas ⁵ ( )

Submited date : 2025-01-17 Accepted date : 2025-03-02

Keywords: Bipolar valued hesitant fuzzy graph, Spectrum, Energy, Laplacian energy.,

Abstract :

Bipolar-valued hesitant fuzzy graphs (BVHFGs) provide a suitable framework for representing knowl- edge in situations characterized by uncertainty, imprecision, and hesitations. Current research shows a lack of studies on energy in contexts involving bipolarity, hesitations, and fuzzy data, motivating us to propose new defi- nitions of energy in this area. In this study, we introduce innovative notions of graph energy and Laplacian energy within the framework of a bipolar valued hesitant fuzzy setting and scrutinize certain characteristics and various types of bounds of these concepts. Additionally, the investigation explores the interplay between the energy and Laplacian energy of BVHFGs. Consequently, a numerical illustration is provided, encompassing the identification of optimal alternatives to elucidate the pragmatic application of the proposed theoretical frameworks within the realm of decision-making. This empirical demonstration underscores the efficacy and relevance of the developed methodology in addressing real-world decision-making challenges.

References:

[1] Zadeh LA, George JK, Bo Y. Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers by Lotfi A Zadeh . Washington: World scientific; 1996. DOI: https://doi.org/10.1142/2895
[2] Allahviranloo T, Abbasbandy S, Rouhparvar H. The exact solutions of fuzzy wave-like equations with variable coefficients by a variational iteration method. Applied Soft Computing . 2011; 11(2): 2186-2192. DOI: https://doi.org/10.1016/j.asoc.2010.07.018
[3] Moloudzadeh S, Allahviranloo T, Darabi P. A new method for solving an arbitrary fully fuzzy linear system. Soft Computing . 2013; 17(9): 1725-1731. DOI: https://doi.org/10.1007/s00500-013-0986-x
[4] Allahviranloo T, Hosseinzadeh Lotfi F, Khorasani Kiasar M, Khezerloo M. On the fuzzy solu- tion of LR fuzzy linear systems. Applied Mathematical Modelling . 2013; 37(3): 1170-1176. DOI: https://doi.org/10.1016/j.apm.2012.03.037
[5] Zhang W-R. Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. NAFIPS/IFIS/NASA’94. Proceedings of the first international joint con- ference of the north American fuzzy information processing society biannual conference. The industrial
fuzzy Control and intelligent, 18-21 December 1994, San Antonio, TX, USA . San Antonio: IEEE; 1994. DOI: https://doi.org/10.1109/IJCF.1994.375115
[6] Mandal P, Ranadive AS. Hesitant bipolar-valued fuzzy sets and bipolar-valued hesitant fuzzy sets and their applications in multi-attribute group decision making. Granular Computing . 2019; 4(3): 559-583. DOI: https://doi.org/10.1007/s41066-018-0118-1
[7] Ali G, Afzal A, Sheikh U, Nabeel M. Multi-criteria group decision-making based on the combination of dual hesitant fuzzy sets with soft expert sets for the prediction of a local election scenario. Granular computing . 2023; 8.6: 2039-2066. DOI: https://doi.org/10.1007/s41066-023-00414-w
[8] Pandey SD, Ranadive AS, Samanta S. Bipolar-valued hesitant fuzzy graph and its application. Social Network Analysis and Mining . 2022; 12(14). DOI: https://doi.org/10.1007/s13278-021-00824-1
[9] Mandal P, Samanta S, Pal M, Ranadive AS. Pythagorean linguistic preference relations and their applications to group decision making using group recommendations based on consistency matrices and feedback mechanism. International Journal of Intelligent Systems .2020; 35(5): 826-849. DOI:
https://doi.org/10.1002/int.22226
[10] Bai W, Ding J, Zhang Ch. Dual hesitant fuzzy graphs with applications to multi-attribute deci- sion making. International Journal of Cognitive Computing in Engineering . 2020; 1: 18-26. DOI: https://doi.org/10.1016/j.ijcce.2020.09.002
[11] Zhang C, Li D, Liang J, et al. MAGDM-oriented dual hesitant fuzzy multigranulation probabilistic models based on MULTIMOORA. International Journal of Machine Learning and Cybernetics . 2021; 12: 1219-1241. DOI: https://doi.org/10.1007/s13042-020-01230-3
[12] Rosenfeld A. In Fuzzy sets and their applications to cognitive and decision processes. Academic Press. 1975: 77-95. DOI: https://www.sciencedirect.com/science/article/pii/B9780127752600500086
[13] Kauffman A. Introduction a la theorie des sousemsembles flous. Paris: Masson; 1973.
[14] Rupkumar M, Samanta S, et al. Colouring of COVID-19 affected region based on fuzzy directed graphs. Computers, Materials & Continua . 2021; 68(1): 1219-1233. DOI: https://doi.org/10.32604/cmc.2021.015590
[15] Pandey SD, Ranadive AS, Samanta S, Sarkar B. BipolarValued Fuzzy Social Network and Centrality Measures. Discrete Dynamics in Nature and Society . 2022; 2022: 1-13. DOI: https://doi.org/10.1155/2022/9713575
[16] Pandey SD, Ranadive AS, et al. RETRACTED: A study on coopetition using bipolar fuzzy bunch graphs. Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology . 2024; 47(1): 1-20. DOI: https://doi.org/10.3233/JIFS-234061
[17] Akram M. Bipolar fuzzy graphs with applications. Knowledge-Based Systems . 2013; 39: 1-8. DOI: https://doi.org/10.1016/j.knosys.2012.08.022
[18] Pathinathan TJ, Arockiaraj J, Jesintha Rosline J. Hesitancy fuzzy graphs. Indian Journal of Science and Technology . 2015; 8(35): 1-5. DOI: https://doi.org/10.17485/ijst/2015/v8i35/86672
[19] Atanassov KT. Review and new results on intuitionistic fuzzy sets. International Journal Bioautomotion . 2016; 20(S1): S7-S16.
[20] Karaaslan F. Hesitant fuzzy graphs and their applications in decision making. Journal of Intelli- gent & Fuzzy Systems: Applications in Engineering and Technology . 2019; 36(3): 2729-2741. DOI: https://doi.org/10.3233/JIFS-18865
[21] Torra V. Hesitant fuzzy sets. International journal of intelligent systems . 2010; 25(6): 529-539. DOI: https://doi.org/10.1002/int.20418
[22] Anjali N, Sunil M. Energy of a fuzzy graph. Annals of fuzzy mathematics and Informatics . 2013; 6(3): 455-465.
[23] Gutman I. The energy of a graph. Berichte der MathematischStatistischen Sektion im Forschungszentrum Graz . 1987; 103: 1-22.
[24] Gutman I. The energy of a graph: old and new results. Algebraic Combinatorics and Applications: Proceedings of the Euroconference, Algebraic Combinatorics and Applications (ALCOMA), held in Gwe- instein, Germany, September 1219, 1999. Berlin: Springer Berlin Heidelberg; 2001: 196-211. DOI:
https://doi.org/10.1007/978-3-642-59448-9 13
[25] Liu H, Lu M, Tian F. Some upper bounds for the energy of graphs. Journal of Mathematical Chemistry . 2007; 41: 45-57. DOI: https://doi.org/10.1007/s10910-006-9183-9
[26] Gutman I, Zhou B. Laplacian energy of a graph. Linear Algebra and its applications . 2006; 414(1): 29-37. DOI: https://doi.org/10.1016/j.laa.2005.09.008
[27] Balakrishnan R. The energy of a graph. Linear Algebra and its Applications . 2004; 387: 287-295. DOI: https://doi.org/10.1016/j.laa.2004.02.038
[28] Cvetkovic D. Applications of graph spectra: an introduction to the literature. Zbornik Radova . 2009; 21: 7-32.
[29] Sasipriya AS, Hemant K. Energy of Fuzzy, Intuitionistic Fuzzy, and Neutrosophic Graphs in Decision Making-A Literature Review. International Journal of Neutrosophic Science (IJNS) . 2025; 25(1): 247- 257. DOI: https://doi.org/10.54216/IJNS.250123
[30] Gutman I, Shao J-Y. The energy change of weighted graphs. Linear algebra and its applications . 2011; 435(10): 2425-2431. DOI: https://doi.org/10.1016/j.laa.2011.02.045
[31] Germina KA, Shahul Hameed K, Zaslavsky T. On products and line graphs of signed graphs, their eigenvalues and energy. Linear Algebra and its Applications . 2011; 435(10): 2432-2450. DOI: https://doi.org/10.1016/j.laa.2010.10.026
[32] Naz S, Ashraf S, Karaaslan F. Energy of a bipolar fuzzy graph and its application in decision making. Italian Journal of Pure and Applied Mathematics . 2018; 40: 339-352.
[33] Rahimi Sharbaf S, Fayazi F. Laplacian energy of a fuzzy graph. Iranian Journal of Mathematical Chem- istry . 2014; 5(1): 1-10. DOI: http://doi.org/10.22052/ijmc.2014.5214
[34] Nithyanandham D, Augustin F, Micheal DR, Pillai ND. Energy based bipolar intuitionistic fuzzy digraph decision-making system in selecting COVID-19 vaccines. Applied Soft Computing . 2023; 147: 110793. DOI: https://doi.org/10.1016/j.asoc.2023.110793
[35] Shao JY, Gong F, Du ZB. Extremal energies of weighted trees and forests with fixed total weight sum. MATCH - Communications in Mathematical and in Computer Chemistry . 2011; 66(3): 879-890.
[36] Riordan O. Spanning subgraphs of random graphs. Combinatorics, Probability and Computing . 2000; 9(2): 125-148. DOI: https://doi.org/10.1017/S0963548399004150
[37] Singh S, Sharma S, Lalotra S. Generalized correlation coefficients of intuitionistic fuzzy sets with appli- cation to MAGDM and clustering analysis. International Journal of Fuzzy Systems . 2020; 22: 1582-1595. DOI: https://doi.org/10.1007/s40815-020-00866-1
[38] Mordeson JN, Nair PS. Fuzzy graphs and fuzzy hypergraphs . Berlin: Physica Heidelberg; 2012. DOI: https://doi.org/10.1007/978-3-7908-1854-3
[39] Fan, K. On a theorem of Weyl concerning eigenvalues of linear transformations I. Proceedings of the National Academy of Sciences USA . 1949; 35(11): 652-655. DOI: https://doi.org/10.1073/pnas.35.11.652
[40] Rajagopal Reddy N, Sharief Basha S. The correlation coefficient of hesitancy fuzzy graphs in decision making. Computer Systems Science and Engineering . 2023; 46(1): 579-596. DOI: https://doi.org/10.32604/csse.2023.034527