Embedding b-Metric Spaces of Reducible Fuzzy Digraphs into Normed Spaces

Umilkeram Qasim  Obaid

doi:10.6180/jase.202602_29(2).0016

Embedding b-Metric Spaces of Reducible Fuzzy Digraphs into Normed Spaces

Mathematics / Statistics

Umilkeram Qasim ObaidThis email address is being protected from spambots. You need JavaScript enabled to view it.

Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq.

Received: October 12, 2024
Accepted: April 10, 2025
Publication Date: June 8, 2025

Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Download Citation: ||https://doi.org/10.6180/jase.202602_29(2).0016

In traditional and iterative algebraic techniques, representing fuzzy digraphs using their adjacency matrices can pose challenges, particularly when dealing with graphs featuring an extensive quantity of nodes or edges. These difficulties are especially pronounced in graphs that do not necessarily exhibit strong connectivity between nodes, which are known as reducible digraphs. In this study, we explore different structure of reducible fuzzy digraphs, employing an innovative b-metric structure. This structure enhances the embedding of massive data entities or nodes into low-dimensional realm depicted by a normed space. Therefore, our aim is to define a novel notion of a distance function called b_RD-distance between any two vertices (or nodes) in reducible fuzzy digraphs and utilize it to introduce quasi-pseudo- b-metric spaces for these digraphs. Furthermore, to minimize b_RD-distance calculations, we demonstrate the process of embedding such b_RD-metrics on reducible fuzzy graphs into the designated normed space ℓ_∞. Ultimately, a computational example, real-world application, and comparative evaluation confirm the practicality of the suggested technique in handling reducibility, asymmetry, and vagueness in fuzzy digraphs.

Keywords: bRD-Distance; Quasi-pseudo- bRD-metric; Embedding bRD-metric into normed space; Fuzzy graph; Reducible graph.

[1] F. Bonomo-Braberman, N. Brettell, A. Munaro, and D. Paulusma, (2024) “Solving problems on generalized convex graphs viamim-width" Journal of Computer and System Sciences 140: 103493. DOI: 10.1016/j.jcss.2023.103493.
[2] G. Jayabalasamy, C. Pujol, and K. Latha Bhaskaran, (2024) “Application of graph theory for blockchain technologies" Mathematics 12(8): 1133. DOI: 10.3390/math12081133.
[3] S. M. Kosti´c, M. I. Simi´c, and M. V. Kosti´c, (2020) “Social network analysis and churn prediction in telecommunications using graph theory" Entropy 22(7): 753. DOI: 10.3390/e22070753.
[4] M. R. Davahli, W. Karwowski, K. Fiok, A. Murata, N. Sapkota, F. V. Farahani, A. Al-Juaid, T. Marek, and R. Taiar, (2022) “The COVID-19 infection diffusion in the US and Japan: A graph-theoretical approach" Biology 11(1): 125. DOI: 10.3390/biology11010125.
[5] S. Brezovnik, N. Tratnik, and P. Žigert Pleteršek, (2021) “Weighted Wiener indices of molecular graphs with application to alkenes and alkadienes" Mathematics 9(2): 153. DOI: 10.3390/math9020153.
[6] N. Saleem, R. Shafqat, R. George, A. Hussain, and M. Yaseen, (2023) “A theoretical analysis on the fractional fuzzy controlled evolution equation" Fractals 31(10): 2340090. DOI: 10.1142/S0218348X2340090X.
[7] K. Abuasbeh and R. Shafqat, (2022) “Fractional Brownian motion for a system of fuzzy fractional stochastic differential equation" Journal of Mathematics 2022(1): 3559035. DOI: 10.1155/2022/3559035.
[8] K. Abuasbeh, R. Shafqat, A. U. K. Niazi, and M. Awadalla, (2022) “Oscillatory behavior of solution for fractional order fuzzy neutral predator-prey system" AIMS Mathematics 7(11): 20383–20400. DOI: 10.3934/math.20221117.
[9] K. Abuasbeh, R. Shafqat, A. U. K. Niazi, and M. Awadalla, (2022) “Local and global existence and unique ness of solution for class of fuzzy fractional functional evolution equation" Journal of Function Spaces 2022(1): 7512754. DOI: 10.1155/2022/7512754.
[10] A. U. K. Niazi, J. He, R. Shafqat, and B. Ahmed, (2021) “Existence, uniqueness, and Eq–Ulam-type stability of fuzzy fractional differential equation" Fractal and Fractional 5(3): 66. DOI: 10.3390/fractalfract5030066.
[11] M. Khan, S. Ullah, M. Zeeshan, R. Shafqat, I. Kebaili, T. B. Bedada, and S. Anis, (2024) “Novel complex fuzzy distance measures with hesitance values and their applications in complex decision-making problems" Scientific Reports 14(1): 14243. DOI: 10.1038/s41598-024-64112-6.
[12] M. Zeeshan, M. Khan, R. Shafqat, A. Althobaiti, S. Althobaiti, and T. B. Bedada, (2024) “Novel similarity measures under complex pythagorean fuzzy soft matrices and their application in decision making problems" Scientific Reports 14(1): 17129. DOI: 10.1038/s41598-024 65324-6.
[13] S.A.Kauffman,(1971)“Cellular homeostasis, epigenesis and replication in randomly aggregated macromolecular systems" Journal of Cybernetics 1(1): 71–96. DOI: 10.1080/01969727108545830.
[14] A. Rosenfeld. “Fuzzy graphs”. In: Fuzzy sets and their applications to cognitive and decision processes. Elsevier, 1975, 77–95. DOI: 10.1016/B978-0-12-775260-0.50008-6.
[15] M. Pal, S. Samanta, and G. Ghorai. Modern trends in fuzzy graph theory. Springer, 2020. DOI: 10.1007/978 981-15-8803-7.
[16] M.Binu,S.Mathew,andJ.N.Mordeson,(2021)“Connectivity status of fuzzy graphs" Information sciences 573: 382–395. DOI: 10.1016/j.ins.2021.05.068.
[17] U. Ahmad and I. Nawaz, (2022) “Directed rough fuzzy graph with application to trade networking" Computational and Applied Mathematics 41(8): 366. DOI: 10.1007/s40314-022-02073-0.
[18] X. Shi, S. Kosari, A. A. Talebi, S. H. Sadati, and H. Rashmanlou, (2022) “Investigation of the main energies of picture fuzzy graph and its applications" Interna tional Journal of Computational Intelligence Systems 15(1): 31. DOI: 10.1007/s44196-022-00086-5.
[19] S. Shanmugam, T. P. Aishwarya, and N. Shreya, (2023) “Bridge domination in fuzzy graphs" Journal of fuzzy extension and applications 4(3): 148–154. DOI: 10.22105/jfea.2023.382170.1251.
[20] J. Scott and M. Tma. Algorithms for sparse linear systems. Springer Nature, 2023. DOI: 10.1007/978-3-031 25820-6.
[21] K. Hayashi, S. G. Aksoy, and H. Park. “Skew symmetric adjacency matrices for clustering directed graphs”. In: 2022 IEEE International Conference on Big Data (Big Data). IEEE. 2022, 555–564. DOI: 10.1109/BigData55660.2022.10020413.
[22] J. Bang-Jensen and G. Z. Gutin. Digraphs: theory, algorithms and applications. Springer Science & Business Media, 2008.
[23] V. Noferini and M. C. Quintana, (2024) “Generating functions of non-backtracking walks on weighted digraphs: Radius of convergence and Ihara’s theorem" Linear Alge bra and its Applications 699: 72–106. DOI: 10.1016/j.laa.2024.06.022.
[24] O. U. Qasim, (2022) “Structures of Normed Path-Edge Spaces of Irreducible Fuzzy Graphs" 18(05): 1645. DOI: 10.24507/ijicic.18.05.1645.
[25] D. Bravo, F. Cubría, M. Fiori, and V. Trevisan, (2023) “Characterization of digraphs with three complementarity eigenvalues" Journal of Algebraic Combinatorics 57(4): 1173–1193. DOI: 10.1007/s10801-023-01218-6.
[26] D. Grinberg, (2023) “An introduction to graph theory" arXiv preprint arXiv:2308.04512: DOI: 10.48550/arXiv.2308.04512.
[27] D. Nithyanandham, F. Augustin, D. R. Micheal, and N. D. Pillai, (2023) “Energy based bipolar intuitionis tic fuzzy digraph decision-making system in selecting COVID-19 vaccines" Applied Soft Computing 147: 110793. DOI: 10.1016/j.asoc.2023.110793.
[28] D. Nithyan and ham and F. Augustin, (2023) “A bipolar fuzzy p-competition graph based ARAS technique for prioritizing COVID-19 vaccines" Applied Soft Computing 146: 110632. DOI: 10.1016/j.asoc.2023.110632.
[29] D. Nithyan and ham, F. Augustin, S. Narayanamoor thy, A. Ahmadian, D. Balaenu, and D. Kang, (2023) “Bipolar intuitionistic fuzzy graph based decision-making model to identify flood vulnerable region" Environmental Science and Pollution Research 30(60): 125254 125274. DOI: 10.1007/s11356-023-27548-3.
[30] A. Banerjee and S. Amanathulla, (2024) “Optimization of disaster management using split domination in picture fuzzy graphs" Journal of Applied Mathematics and Computing 70(1): 435–459. DOI: 10.1007/s12190-023-01965-6.
[31] S. R. Islam and M. Pal, (2023) “Further development of F-index for fuzzy graph and its application in Indian railway crime" Journal of applied mathematics and computing 69(1): 321–353. DOI: 10.1007/s12190-022 01748-5.
[32] S. R. Islam and M. Pal, (2023) “An investigation of edge F-index on fuzzy graphs and application in molecu lar chemistry" Complex & Intelligent Systems 9(2): 2043–2063. DOI: 10.1007/s40747-022-00896-2.
[33] S. Broumi, M. Talea, A. Bakali, F. Smar and ache, D. Nagarajan, M. Lathamaheswari, and M. Parimala, (2019) “Shortest path problem in fuzzy, intuitionistic fuzzy and neutrosophic environment: an overview" Com plex & Intelligent Systems 5: 371–378. DOI: 10.1007/ s40747-019-0098-z.
[34] S. Halder, S. Majumder, A. Biswas, B. K. Mandal, and S.-L. Peng. “On Multi-objective Fuzzy Shortest Path Problem”. In: International Conference on Advances in Data Science and Computing Technologies. Springer. 2022, 499–509. DOI: 10.1007/978-981-99-3656-4_51.
[35] J. N. Mordeson and S. Mathew. Advanced topics in fuzzy graph theory. 375. Springer, 2019. DOI: 10.1007/ 978-3-030-04215-8.
[36] E. Enriquez, G. Estrada, C. Loquias, R. J. Bacalso, and L. Ocampo, (2021) “Domination in fuzzy directed graphs" Mathematics 9(17): 2143. DOI: 10.3390/math9172143.
[37] M. Tom and M. S. Sunitha,(2015)“Strongsumdistance in fuzzy graphs" Springer Plus 4: 1–14. DOI: 10.1186/s40064-015-0935-5.
[38] S. D. Mohsen and Y. H. Thiyab, (2024) “Some Characteristics Of Completeness Property In Fuzzy Soft b-Metric Space" Journal of Applied Science and Engineering 27(3): 2227–2232. DOI: 10.6180/jase.202403_27(3).0009.
[39] S. Nadaban, (2023) “Fuzzy quasi-b-metric spaces" Ann. West Univ. Timis. Math. Comput. Sci 8: 38–48. DOI: 10.2478/awutm-2022-0015.
[40] S. N˘ad˘aban, (2016) “Fuzzy b-metric spaces" International Journal of Computers Communications & Control 11(2): 273–281. DOI: 10.15837/ijccc.2016.2.2443.
[41] D. Doli´ canin-Ðeki´ c, T. Došenovi´c, H. Huang, and S. Radenovi´c, (2016) “A note on recent cyclic fixed point results in dislocated quasi-b-metric spaces" Fixed Point Theory and Applications 2016: 1–10. DOI: 10.1186/s13663-016-0565-9.
[42] N. Zikria, M. Samreen, T. Kamran, H. Aydi, and C. Park, (2022) “Some periodic and fixed point theorems on quasi-b-gauge spaces" Journal of Inequalities and Applications 2022(1): 13. DOI: 10.1186/s13660-021 02750-4.
[43] J. Matoušek. Lecture notes on metric embeddings. Tech. rep. Technical report, ETH Zürich, 2013.
[44] D. Taha, W. Zhao, J. M. Riestenberg, and M. Strube, (2023) “Normed Spaces for Graph Embedding" arXiv preprint arXiv:2312.01502: DOI: 10.48550/arXiv.2312.01502.
[45] A. Gu, F. Sala, B. Gunel, and C. Ré. “Learning mixed curvature representations in product spaces”. In: In ternational conference on learning representations. 2018.
[46] F. Lopez, B. Pozzetti, S. Trettel, M. Strube, and A. Wienhard. “Symmetric spaces for graph embeddings: Afinsler-riemannian approach”. In: International Con ference on Machine Learning. PMLR. 2021, 7090–7101.
[47] F. Di Giovanni, G. Luise, and M. Bronstein, (2022) “Heterogeneous manifolds for curvature-aware graphem bedding" arXiv preprint arXiv:2202.01185: DOI: 10.48550/arXiv.2202.01185.
[48] J. Matoušek, (1996) “On the distortion required for embedding finite metric spaces into normed spaces" Israel Journal of Mathematics 93(1): 333–344. DOI: 10.1007/BF02761110.
[49] J. Matousek. Lectures on discrete geometry. 212. Springer Science & Business Media, 2013. DOI: 10. 1007/978-1-4613-0039-7.