INTRODUCTION TO Q-FRACTIONAL FUZZY GRAPHS AND THEIR ENERGY ANALYSIS

Document Type : Research Paper

Authors

1 COMSATS University Islamabad Abbottabad Campus

2 COMSATS University Islamabad, Abbottabad Campus.

Abstract

Over the years, traditional fuzzy graph models such as Intuitionistic Fuzzy Graphs (IFGs), Pythagorean Fuzzy Graphs (PFGs), and q-Rung Orthopair Fuzzy Graphs (q-ROFGs) have emerged as powerful tools for modeling uncertainty in complex networked structures. Despite their advancements, these models exhibit critical limitations, particularly in scenarios where the sum of membership and non-membership degrees exceeds one. For example, graph vertex data of the form {(v₁,(1,0.5)),(v₂,(1,1)),(v₃,(0.3,1))} or edge data of the form {(e₁,(1,1)),(e₂,(1,0.6)),(e₃,(0.3,1))} violate the constraint 0≤μ+ν≤1. Moreover, existing models often struggle to eliminate the inherent dependency between the membership and non-membership grades.
To overcome these limitations, we propose a novel framework called the q-Fractional Fuzzy Graph (q-FFG), which extends fuzzy graph theory, allows greater flexibility and independence between membership and non-membership grades. Unlike previous models, q-fractional fuzzy graphs can handle extreme degrees of uncertainty by letting both membership and non-membership values approach 1 independently, without violating consistency. We formally define the structure of q-fractional fuzzy graphs and some operations on these graphs. Additionally, we discuss degree and adjacency matrices of these graphs. These fundamental matrices serve as essential tools for understanding the structure and behavior of these graphs. Through illustrative examples, we highlight their practical applications and significance. The study presents an extensive review of the Laplacian matrices and their associated energies, uncovering essential information about the graph's spectral properties. These mathematical formulations are necessary for analyzing the stability and connectivity of q-fractional fuzzy graphs. Additionally, an algorithm based on Principal Component Analysis (PCA) technique is proposed which effectively reduces the dimensionality of data while preserving the essential features.

Keywords

Main Subjects

References

[1] M. Akram, K. Zahid, J. C. R Alcantud, A new outranking method for multicriteria decision making with complex
Pythagorean fuzzy information, Neural Computing and Applications, 34(10) (2022), 8069-8102. https://doi.org/
10.1007/s00521-021-06847-1
[2] J. C. R. Alcantud, N. Stojanovi´c, L. Djurovi´c, M. Lakovi´c, Decision-making and clustering algorithms based on the
scored-energy of hesitant fuzzy soft sets, International Journal of Fuzzy Systems, (2025), 1-15. https://doi.org/
10.1007/s40815-024-01964-0
[3] H. Alohali, M. B. Khan, J. E. Mac´ıas-D´ıaz, F. Sikander, On (p,q)-fractional linear Diophantine fuzzy sets and their
applications via MADM approach, AIMS Mathematics, 9(12) (2024), 35503-35532. https://doi.org/10.3934/
math.20241685
[4] Z. A. Ameen, H. F. M. Salih, W. Alagal, B. A. Asaad, q-fractional hesitant fuzzy multicriteria decision-making
approaches for uncertain power system performance evaluation, Energy Science and Engineering, (2025). https:
//doi.org/10.1002/ese3.70242
[5] K. T. Atanassov, Intuitionistic fuzzy sets, Springer, Heidelberg, 1999. https://doi.org/10.1007/
978-3-7908-1870-3
[6] K. T. Atanassov, On intuitionistic fuzzy sets theory, Springer-Verlag: Heidelberg, 2012. https://doi.org/10.
1007/978-3-642-29127-2
[7] K. T. Atanassov, G. Pasi, R. Yager, V. Atanassova, Intuitionistic fuzzy graph interpretations of multi-person multicriteria
decision making, Conference: Proceedings of the 3rd Conference of the European Society for Fuzzy Logic
and Technology, Zittau, Germany, (2003).
[8] S. S. Basha, S. Sharief, E. Kartheek, Laplacian energy of an intuitionistic fuzzy graph, Indian Journal of Science
and Technology, 8(33) (2015), 1-7. https://doi.org/10.17485/ijst/2015/v8i33/79899
[9] L. Djurovi´c, M. Lakovi´c, N. Stojanovi´c, Decision-making algorithm based on the energy of interval-valued fuzzy
soft sets, arXiv preprint arXiv:2405.15801, (2024). https://doi.org/10.48550/arXiv.2405.15801
[10] M. Gulistan, W. Pedrycz, Introduction to q-fractional fuzzy set, International Journal of Fuzzy System, 26(5)
(2024), 1399-1416. https://doi.org/10.1007/s40815-023-01633-8
[11] I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra and its Applications, 414(1) (2006), 29-37.
https://doi.org/10.1016/j.laa.2005.09.008
[12] A. Habib, M. Akram, A. Farooq, q-rung orthopair fuzzy competition graphs with application in the soil ecosystem,
Mathematics, 7(1) (2019). https://doi.org/10.3390/math7010091
[13] S. Khan, M. I. Syam, H. A. Abujabal, M. Rahim, A. Alburaikan, H. A. E. W. Khalifa, A strategic decision-making
framework for evaluating barriers to green supply chain management using fractional fuzzy similarity measures,
Scientific Reports, 15(1) (2025), 29089. https://doi.org/10.1038/s41598-025-13963-8
[14] L. Mudri´c-Staniˇskovski, L. Djurovic, N. Stojanovi´c, Energy of a fuzzy soft set and its application in decision-making,
Iranian Journal of Fuzzy Systems, 21(2) (2024), 35-49. https://doi.org/10.22111/ijfs.2024.46797.8243
[15] A. Narayanan, S. Mathew, Energy of a fuzzy graph, Annals of Fuzzy Mathematics and Informatics, 6(3) (2013),
455-465. http://www.kyungmoon.com
[16] S. Naz, S. Ashraf, M. Akram, A novel approach to decision-making with Pythagorean fuzzy information, Mathematics,
(2018), 1-28. https://doi.org/10.3390/math6060095
[17] N. Stojanovi´c, M. Lakovi´c, L. Djurovi´c, Decision-making algorithm based on the energy of interval-valued hesitant
fuzzy soft sets, Neural Computing and Applications, 37(16) (2025), 9821-9841. https://doi.org/10.1007/
s00521-025-11107-7
[18] R. R. Yager, Pythagorean fuzzy subsets, Joint IFSAWorld Congress and UAFIPS Annual Meeting (IFSA/uAFIPS),
(2013), 57-61. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375
[19] R. R. Yager, Generalized orthopair fuzzy sets, IEEE Transactions on Fuzzy System, 25(5) (2016), 1222-1230.
http://doi.org/10.1109/TFUZZ.2016.2604005
[20] L. A. Zadeh, Fuzzy set, Information Control, 8(3) (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)
90241-X
Iranian Journal of Fuzzy Systems
Volume 22, Issue 6
November and December 2025
Pages 83-101

Files

Share

How to cite

Statistics