Homomorphic and restricted homomorphic products of soft graphs

Authors

  • Jinta Jose
    Department of Science and Humanities, Viswajyothi College of Engineering and Technology Vazhakulam, India
  • Rajesh K. Thumbakara
    Department of Mathematics, Mar Athanasius College (Autonomous) Kothamangalam, India
  • J. D. Thenge Mashale
    School of Computational Sciences, Punyashlok Ahilyadevi Holkar Solapur University, Solapur, India
  • Bobin George
    Department of Mathematics Pavanatma College Murickassery, India
  • Sijo P. George
    Department of Mathematics Pavanatma College Murickassery, India

Keywords:

Soft Set, Soft Graph, Homomorphic Product, Restricted homomorphic products

Abstract

Molodtsov pioneered the notion of soft set theory, presenting it as a mathematical tool for dealing with uncertainty. Numerous researchers have subsequently developed models leveraging this theory to tackle challenges in decision-making and medical diagnosis. Soft set theory emerges as a flexible framework adept at handling uncertain and imprecise information, a domain where classical set theory often struggles. Expanding on the soft set concept, researchers have introduced the idea of a soft graph. This innovative concept allows for the creation of diverse representations of graph-based relations by incorporating parameterisation. In this work, we present and investigate some of the features of the homomorphic and restricted homomorphic products of soft graphs. This paper establishes the structural properties of these products, ensuring that they are well-defined and maintain the essential characteristics of soft graphs. Additionally, we derive combinatorial identities related to the counts of vertices and edges, as well as the degree sums, offering deeper insights into the composition and behaviour of these graph products.

Dimensions

[1] D. Molodtsov, “Soft set theory-first results”, Computers & Mathematics with Applications 37 (1999) 19. https://doi.org/10.1016/S0898-1221(99)00056-5.

[2] P. K. Maji & A. R. Roy, “A fuzzy soft set theoretic approach to decision-making problems”, Journal of Computational and Applied Mathematics 203 (2007) 412. https://doi.org/10.1016/j.cam.2006.04.008.

[3] P. K. Maji, A. R. Roy & R. Biswas, “An application of soft sets in a decision making problem”, Computers and Mathematics with Application 44 (2002) 1077. https://doi.org/10.1016/S0898-1221(02)00216-X.

[4] S. S. Mohammed, “On fuzzy soft set-valued maps with application”, Journal of the Nigerian Society of Physical Sciences 2 (2020) 26. https://doi.org/10.46481/jnsps.2020.48.

[5] S. Saleh, T. M. Al-Shami, L. R. Flaih, M. Arar & R. Abu-Gdairi, “Ri-Separation axioms via supra soft topological spaces”, J. Math. Compt. Sci. 32 (2024) 263. http://dx.doi.org/10.22436/jmcs.032.03.07.

[6] S. Saleh, L. R. Flaih & K. F. Jasim, “Some applications of soft ?-closed sets in soft closure space”, Communications in Mathematics and Applications 14 (2023) 481. https://www.researchgate.net/publication/373629819_Some_applications_of_soft_-closed_sets_in_soft_closure_spaces.

[7] R. K. Thumbakara & B. George, “Soft graphs”, Gen. Math. Notes 21 (2014) 75. http://emis.icm.edu.pl/journals/GMN/yahoo_site_admin/assets/docs/6_GMN-4802-V21N2.16902935.pdf.

[8] R. K. Thumbakara, B. George & J. Jose, “Subdivision graph, power and line graph of a soft graph”, Communications in Mathematics and Applications 13 (2022) 75. https://doi.org/10.26713/cma.v13i1.1669.

[9] M. Akram & S. Nawaz, “Operations on soft graphs”, Fuzzy Inf. Eng. 7 (2015) 423. https://doi.org/10.1016/j.fiae.2015.11.003.

[10] M. Akram & S. Nawaz, “Certain types of soft graphs”, U. P. B. Sci. Bull., Series A 78 (2016) 67. https://www.scientificbulletin.upb.ro/rev_docs_arhiva/fullcc2_842873.pdf.

[11] M. Akram & S. Nawaz, “On fuzzy soft graphs”, Italian Journal of Pure and Applied Mathematics 34 (2015) 463. https://ijpam.uniud.it/online_issue/201534/44-AkramNawaz.pdf.

[12] M. Akram & S. Nawaz, “Fuzzy soft graphs with applications”, Journal of Intelligent and Fuzzy Systems 30 (2016) 3619. https://doi.org/10.3233/IFS-162107.

[13] M. Akram & F. Zafar, “On soft trees”, Buletinul Acad. Stiinte a Republicia Moldova. Mathematica 2 (2015) 82. https://www.math.md/files/basm/y2015-n2/y2015-n2-(pp82-95).pdf.

[14] M. Akram & F. Zafar, “Fuzzy soft trees”, Southeast Asian Bulletin of Mathematics 40 (2016) 151. http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_201602&filename=01_40(2).pdf.

[15] J. D. Thenge, B. S. Reddy & R. S. Jain, “Connected soft graph”, New Mathematics and Natural Computation 16 (2020) 305. https://doi.org/10.1142/S1793005720500180.

[16] J. D. Thenge, B. S. Reddy & R. S. Jain, “Contribution to soft graph and soft tree”, New Mathematics and Natural Computation 15 (2019) 129. https://doi.org/10.1142/S179300571950008X.

[17] J. D. Thenge, B. S. Reddy & R. S. Jain, “Adjacency and incidence matrix of a soft graph”, Communications in Mathematics and Applications 11 (2020) 23. https://doi.org/10.26713/cma.v11i1.1281.

[18] B. George, J. Jose & R. K. Thumbakara, “An introduction to soft hypergraphs”, Journal of Prime Research in Mathematics 18 (2022) 43. https://jprm.sms.edu.pk/an-introduction-to-soft-hypergraphs/.

[19] J. Jose, B. George & R. K. Thumbakara, “Soft directed graphs, their vertex degrees, associated matrices and some product operations”, New Mathematics and Natural Computation 19 (2023) 651. https://doi.org/10.1142/S179300572350028X.

[20] J. Jose, B. George & R. K. Thumbakara, “Soft directed graphs, some of their operations & properties”, New Mathematics and Natural Computation 20 (2024) 129. https://doi.org/10.1142/S1793005724500091.

[21] B. George, R. K. Thumbakara & J. Jose, “Soft disemigraphs, degrees and digraphs associated and their AND & OR operations”, New Mathematics and Natural Computation 20 (2024) 835. https://doi.org/10.1142/S1793005723500448.

[22] R. Hammack, W. Imrich & S. Klavzar, Handbook of product graphs, CRC Press, U.S.A, 2011. https://doi.org/10.1201/b10959.

[23] M. Baghernejad & R. A. Borzooei, “Results on soft graphs and soft multigraphs with application in controlling urban traffic flows”, Soft Computing 27 (2023) 11155. https://doi.org/10.1007/s00500-023-08650-7.

[24] J. Jose, B. George & R. K. Thumbakara, “Eulerian and Unicursal soft graphs”, Mapana Journal of Sciences 22 (2023) 99. https://journals.christuniversity.in/index.php/mapana/article/view/4144.

[25] R. K. Thumbakara, J. Jose & B. George, “Hamiltonian soft graphs”, Ganita 72 (2022) 145. https://bharataganitaparisad.com/wp-content/uploads/2022/11/721-ch15.pdf.

[26] R. K. Thumbakara, J. Jose & B. George, “On soft graph isomorphism”, New Mathematics and Natural Computation, (2023) (Published Online). https://doi.org/10.1142/S1793005725500073.

[27] B. George, J. Jose & R. K. Thumbakara, “Tensor products and strong products of soft graphs”, Discrete Mathematics, Algorithms and Applications 15 (2023) 1. https://doi.org/10.1142/S1793830922501713.

[28] B. George, J. Jose & R. K. Thumbakara, “Co-normal products and modular products of soft graphs”, Discrete Mathematics, Algorithms and Applications 16 (2024) 1. https://doi.org/10.1142/S179383092350012X.

[29] B. George, J. Jose & R. K. Thumbakara, “Modular product of soft directed graphs”, TWMS Journal of Applied and Engineering Mathematics 14 (2024) 966. https://jaem.isikun.edu.tr/web/images/articles/vol.14.no.3/07.pdf.

[30] J. Jose, B. George & R. K. Thumbakara, “Homomorphic product of soft directed graphs”, TWMS Journal of Applied and Engineering Mathematics 14 (2024) 1405. https://jaem.isikun.edu.tr/web/images/articles/vol.14.no.4/07.pdf.

[31] J. Jose, B. George & R. K. Thumbakara, “Rooted product and restricted rooted product of soft directed graphs”, New Mathematics and Natural Computation 20 (2024) 345. https://doi.org/10.1142/S1793005724500194.

[32] J. Jose, B. George & R. K. Thumbakara, “Disjunctive product of soft directed graphs”, New Mathematics and Natural Computation, (2023) (Published Online). https://doi.org/10.1142/S1793005725500139.

[33] J. Jose, B. George & R. K. Thumbakara, “Corona product of soft directed graphs”, New Mathematics and Natural Computation, (2023) (Published Online). https://doi.org/10.1142/S179300572550022X.

[34] J. Jose, B. George, R. K. Thumbakara & Sijo P. George“Understanding normal and restricted normal products in soft directed graphs”, Journal of the Nigerian Society of Physical Sciences 6 (2024) 2156. https://doi.org/10.46481/jnsps.2024.2156.

[35] B. George, J. Jose & R. K. Thumbakara, “Connectedness in soft semigraphs”, New Mathematics and Natural Computation 20 (2024) 157. https://doi.org/10.1142/S1793005724500108.

[36] B. George, R. K. Thumbakara & J. Jose, “Soft semigraphs and some of their operations”, New Mathematics and Natural Computation 19 (2023) 369. https://doi.org/10.1142/S1793005723500126.

[37] B. George, R. K. Thumbakara & J. Jose, “Soft semigraphs and different types of degrees, graphs and matrices associated with them”, Thai Journal of Mathematics 21 (2023) 863. https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1551.

[38] B. George, J. Jose & R. K. Thumbakara, “Investigating the traits of soft semigraph-associated degrees”, New Mathematics and Natural Computation, 20 (2024) 647. http://dx.doi.org/10.1142/S1793005724500352.

[39] B. George, J. Jose & R. K. Thumbakara, “Cartesian product and composition of soft semigraphs”, New Mathematics and Natural Computation, 2023 (Published Online). http://dx.doi.org/10.1142/S1793005725500036.

[40] B. George, R. K. Thumbakara & J. Jose, “Soft directed hypergraphs and their AND & OR operations”, Mathematical Forum 30 (2022) 1. https://mathematical-forum.org/soft-directed-hypergraphs-and-their-and-or-operations/.

Published

2025-02-01

How to Cite

Homomorphic and restricted homomorphic products of soft graphs. (2025). Journal of the Nigerian Society of Physical Sciences, 7(1), 2391. https://doi.org/10.46481/jnsps.2025.2391

Issue

Volume 7, Issue 1, February 2025

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2024 Jinta Jose, Rajesh K. Thumbakara, J. D. Thenge Mashale, Bobin George, Sijo P. George

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Homomorphic and restricted homomorphic products of soft graphs. (2025). Journal of the Nigerian Society of Physical Sciences, 7(1), 2391. https://doi.org/10.46481/jnsps.2025.2391

Similar Articles

31-40 of 88

You may also start an advanced similarity search for this article.

Most read articles by the same author(s)