On the normal product and modular product of product fuzzy graphs with applications to defense communication networks
DOI:
https://doi.org/10.58524/app.sci.def.v4i1.1147Keywords:
Defense Communication Network, Fuzzy Graph, Modular Product , Normal Product , Product Fuzzy GraphAbstract
Background: This study focuses on the development of new operations of product fuzzy graphs, specifically the normal product and modular product operations. The normal product of product fuzzy graphs is used to build infrastructure with maximum path redundancy. Meanwhile, the modular product operation of product fuzzy graphs is implemented to optimize the alignment of operational processes and information security.
Aims: This study aims to define the normal product and modular product of product fuzzy graphs, define the properties of the normal product and modular product of product fuzzy graphs, and to apply to defense communication networks.
Method: The research stages consist of formulating the definition of the normal product and modular product of product fuzzy graphs along with its properties and applying to defense communication networks.
Results: The results of the study show that the normal product and modular product of product fuzzy graph is fuzzy graph, the normal product and modular product of strong product fuzzy graph is strong, consequently, the normal product and modular product of complete product fuzzy graph is strong.
Conclusion: The application of the normal product and modular product of product fuzzy graph can produce a more compact defense communication network.
References
Ahmad, U., Khan, N. K., & Saeid, A. B. (2023). Fuzzy topological indices with application to cybercrime problem. Granular Computing, 8(5), 967–980. https://doi.org/10.1007/s41066-023-00365-2
Ahmad, U., Nawaz, I., & Broumi, S. (2023). Connectivity index of directed rough fuzzy graphs and its application in traffic flow network. Granular Computing, 8(6), 1561–1568. https://doi.org/10.1007/s41066-023-00384-z
Akram, M., & Dudek, W. A. (2011). Interval-valued fuzzy graphs. Computers and Mathematics with Applications, 61(2), 289–299. https://doi.org/10.1016/j.camwa.2010.11.004
Akram, M., Habib, A., Ilyas, F., & Mohsan Dar, J. (2018). Specific Types of Pythagorean Fuzzy Graphs and Application to Decision-Making. Mathematical and Computational Applications, 23(3), 1–28. https://doi.org/10.3390/mca23030042
Al-Hawary, T. (2024). Corona Product of Product Fuzzy Graphs. South East Asian Journal of Mathematics and Mathematical Sciences, 20(3), 25–32. https://doi.org/10.56827/SEAJMMS.2024.2003.3
Al-Hawary, T. (2025a). γ-Product of Product Fuzzy Graphs. International Journal of Fuzzy Logic and Intelligent Systems, 25(2), 207–212. https://doi.org/10.5391/IJFIS.2025.25.2.207
Al-Hawary, T. (2025b). α-Product of Anti Product Fuzzy Graphs. International Journal of Fuzzy Logic and Intelligent Systems, 25(1), 82–86. https://doi.org/10.5391/IJFIS.2025.25.1.82
Al-Hawary, T., & Hashim, M. (2025). Semi-Fuzzy Graphs. Boletim Da Sociedade Paranaense de Matematica, 43(3), 1–10. https://doi.org/10.5269/bspm.63005
Al-Hawary, T., & Hourani, B. (2016). On product fuzzy graphs. Annals of Fuzzy Mathematics and Informatics, 12(2), 279–294. http://www.afmi.or.kr
Al-Hawary, T., & Hourani, B. (2017). On intuitionistic product fuzzy graphs. Italian Journal of Pure and Applied Mathematics, 12(38), 113–126. https://ijpam.uniud.it/online_issue/201738/12-Al-Hawary-Hourani.pdf
Al-Omeri, W. F., Kaviyarasu, M., & Venitha, R. (2025). A Degree-Based Exponential Fuzzy Graph for Pollution Impact Analysis. European Journal of Pure and Applied Mathematics, 18(3), 6479. https://doi.org/10.29020/nybg.ejpam.v18i3.6479
Alqahtani, M., Keerthana, R., Venkatesh, S., & Kaviyarasu, M. (2024). Hesitant Bipolar-Valued Intuitionistic Fuzzy Graphs for Identifying the Dominant Person in Social Media Groups. Symmetry, 16(10), 1293. https://doi.org/10.3390/sym16101293
Azeem, M., Rashid, H., Jamil, M. K., Gütmen, S., & Tirkolaee, E. B. (2025). Plithogenic Fuzzy Graph: A Study Of Fundamental Properties And Potential Applications. Journal of Dynamics and Games, 12(3), 196–214. https://doi.org/10.3934/jdg.2024023
Dhanya, C. M., Mathew, S., & Mordeson, J. N. (2025). Connectivity in Cartesian products of fuzzy graphs: application to decision making. Journal of Applied Mathematics and Computing, 71(3), 4169–4195. https://doi.org/10.1007/s12190-025-02389-0
Dogra, S. (2015). Different Types of Product of Fuzzy Graphs. Progress in Nonlinear Dynamics and Chaos, 3(1), 41–56. www.researchmathsci.org
Firmansah, F. (2025). The Development of New Operations for Tensor Products of Product Fuzzy Graphs , Strong Product Fuzzy Graphs , and Complete Product Fuzzy Graphs. Sainmatika: Jurnal Ilmiah Matematika Dan Ilmu Pengetahuan Alam, 22(2), 140–145. https://doi.org/10.31851/sainmatika.v22i2.20347
Firmansah, F., Tasari, T., & Yuwono, M. R. (2025). The development of cartesian product operation of product fuzzy graphs and its properties. Desimal: Jurnal Matematika, 8(2), 207–214. https://doi.org/10.24042/7edag538
Ganesan, B., Raman, S., Marayanagaraj, S., & Broumi, S. (2023). Geodetic domination integrity in fuzzy graphs. Journal of Intelligent and Fuzzy Systems, 45(2). https://doi.org/10.3233/JIFS-223249
Gulistan, M., Yaqoob, N., Rashid, Z., Smarandache, F., & Wahab, H. A. (2018). A study on neutrosophic cubic graphs with real life applications in industries. Symmetry, 10(6), 203. https://doi.org/10.3390/sym10060203
Islam, S. R., & Pal, M. (2023). Second Zagreb index for fuzzy graphs and its application in mathematical chemistry. Iranian Journal of Fuzzy Systems, 20(1). https://doi.org/10.22111/ijfs.2023.7350
Javaid, M., Kashif, A., & Rashid, T. (2020). Hesitant Fuzzy Graphs and Their Products. Fuzzy Information and Engineering, 12(2), 238–252. https://doi.org/10.1080/16168658.2020.1817658
Josen, J., & John, S. J. (2026a). On Connectivity of Multidimensional Fuzzy Graphs and its Applications. New Mathematics and Natural Computation, 22(2), 449–471. https://doi.org/10.1142/S1793005726500237
Josen, J., & John, S. J. (2026b). On Domination and Related Concepts of Multidimensional Fuzzy Graphs and Their Applications. New Mathematics and Natural Computation, 22(2), 473–494. https://doi.org/10.1142/S1793005726500249
Khatun, J., Amanathulla, S., & Pal, M. (2024). Picture fuzzy cubic graphs and their applications. Journal of Intelligent and Fuzzy Systems, 46(1). https://doi.org/10.3233/JIFS-232523
Khatun, J., Bera, B., Amanathulla, S., & Pal, M. (2025). Certain Operations On m-Polar Picture Fuzzy Graphs. Turkish World Mathematical Society Journal of Applied and Engineering Mathematics, 15(4), 821–839. https://jaem.isikun.edu.tr/web/images/articles/vol.15.no.4/05.pdf
Meenakshi, A., Mishra, J. S., Mršić, L., Kalampakas, A., Samanta, S., & Allahviranloo, T. (2026). Maximal product-based intuitionistic fuzzy line graphs for healthcare predictive analysis. Ain Shams Engineering Journal, 17(1), 1–11. https://doi.org/10.1016/j.asej.2025.103939
Nagarajan, D., Lathamaheswari, M., Broumi, S., & Kavikumar, J. (2019). Dombi interval valued neutrosophic graph and its role in traffic control management. Neutrosophic Sets and Systems, 24, 114–133. https://www.researchgate.net/profile/Nagarajan-Deivanayagampillai/publication/331801121_DombiIntervalValuedNeutro/links/5c925b89299bf111693a3b50/DombiIntervalValuedNeutro.pdf
Nair, K. R., & Sunitha, M. S. (2024). Strong incidence domination in some operations of fuzzy incidence graphs and application in security allocation. International Journal of Mathematics for Industry, 16(1), 2450022. https://doi.org/10.1142/S2661335224500229
Nawaz, I., & Ahmad, U. (2023). Certain Concepts in Directed Rough Fuzzy Graphs and Application to Mergers of Companies. Fuzzy Information and Engineering, 15(3), 248–273. https://doi.org/10.26599/FIE.2023.9270019
Nazeer, I., Rashid, T., & Keikha, A. (2021). An Application of Product of Intuitionistic Fuzzy Incidence Graphs in Textile Industry. Complexity, 2021(1), 1–12. https://doi.org/10.1155/2021/5541125
Poonia, M., & Bhardwaj, P. (2023). The Concept of Complex Neutrosophic Hesitant Fuzzy Graph is Used to Solve a Problem Related to Cellular Network. 2023 International Conference on Sustainable Emerging Innovations in Engineering and Technology, ICSEIET 2023. https://doi.org/10.1109/ICSEIET58677.2023.10303510
Raja, J. R., Lee, J. G., Dhotre, D., Mane, P., Rajankar, O. S., Kalampakas, A., Jambhekar, N. D., & Bhalke, D. G. (2024). Fuzzy graphs and their applications in finding the best route, dominant node and influence index in a network under the hesitant bipolar-valued fuzzy environment. Complex and Intelligent Systems, 10(4), 5195–5211. https://doi.org/10.1007/s40747-024-01438-8
Ramaswamy, V., & Poornima, B. (2009). Product Fuzzy Graphs. IJCSNS International Journal of Computer Science and Network Security, 9(1), 114–118. http://paper.ijcsns.org/07_book/200901/20090116.pdf
Rao, Y., Lei, S., Talebi, A. A., & Mojahedfar, M. (2023). A Novel Concept of Level Graph in Interval-Valued Fuzzy Graphs with Application. Symmetry, 15(12), 2106. https://doi.org/10.3390/sym15122106
Rezayi, M., Eslahchi, C., & Borzooei, R. A. (2025). Connectivity number in fuzzy graphs with application in mobile communication networks. Journal of Applied Mathematics and Computing, 71(Suppl 1), 973–999. https://doi.org/10.1007/s12190-025-02505-0
Rosenfeld, A. (1975). Fuzzy Graphs. Fuzzy Sets and Their Applications, 77–95.
Sankar, C., Kalaivani, C., Saravanan, M., & Sujatha, R. (2025). An algorithmic approach to signed fuzzy graph integrity: Complexity, graph operations, and metro rail network applications. Ain Shams Engineering Journal, 16(9), 103509. https://doi.org/10.1016/j.asej.2025.103509
Shubatah, M. M. ., & Haifa, A. A. (2023). The Global Domination Number In Product Fuzzy Graphs. International Journal of Mathematics and Statistics, 6(1), 29–38. https://doi.org/10.53555/eijms.v6i1.45
Sultana, F., Gulistan, M., Ali, M., Yaqoob, N., Khan, M., Rashid, T., & Ahmed, T. (2023). A study of plithogenic graphs: applications in spreading coronavirus disease (COVID-19) globally. Journal of Ambient Intelligence and Humanized Computing, 14(10), 13139–13159. https://doi.org/10.1007/s12652-022-03772-6
Wang, Q., & Ren, H. (2025). On the Edge Connectivity of Semi-strong Product Graphs. Applied and Computational Mathematics, 14(6), 356–359. https://doi.org/10.11648/j.acm.20251406.15
Wang, Y., Talebi, A. A., Sadeghi, M., Khan, A., & Shafi, J. (2025). An Effective Approach in Fuzzy Graph Molecular Modeling Using Randic Index and Its Applications in Medicinal Chemistry for Parkinson’s Drugs. Journal of Mathematics, 2025(1), 1–17. https://doi.org/10.1155/jom/1390464
Zeng, S., Shoaib, M., Ali, S., Abbas, Q., & Nadeem, M. S. (2020). Complex vague graphs and their application in decision-making problems. IEEE Access, 8, 174094–174104. https://doi.org/10.1109/ACCESS.2020.3025974
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Fery Firmansah, Niken Retnowati, Tasari
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
