Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives

Takaaki Fujita; Florentin Smarandache

doi:10.47000/tjmcs.1729279

Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives

Abstract

A hypergraph is a generalization of a graph in which edges—called hyperedges—can connect any number of vertices, not just two. A superhypergraph further extends this framework by introducing recursive powerset constructions, enabling the representation of hierarchical relationships among hyperedges themselves. A topological graph is a graph embedded in the plane, where each vertex is represented by a distinct point and each edge is drawn as a continuous curve connecting its endpoints. A topological hypergraph extends this idea by representing hyperedges as closed curves that enclose sets of vertices on the plane [13]. In this paper, we introduce the concept of a Topological n-SuperHypergraph, which unifies the structural hierarchy of n-superhypergraphs with the geometric intuition of topological hypergraphs. We anticipate that this new formulation will contribute to future developments in graph theory, topology, and network theory.

Keywords

Supporting Institution

This research received no financial support from any external agencies or organizations.

Ethical Statement

The authors declare that there are no competing interests related to this manuscript.

Thanks

The authors gratefully acknowledge the support, insights, and encouragement provided by colleagues, mentors, and reviewers. We appreciate the interest of our readers and the foundational work of scholars whose publications informed this study. We also thank the institutions and individuals who provided essential resources and infrastructure. Finally, we are indebted to everyone who offered assistance in any form.

References

Ackerman, E., On topological graphs with at most four crossings per edge, Computational Geometry, 85(2019), 101574.
Ackerman, E., Fox, J., Pach, J., Suk, A., On grids in topological graphs, Computational Geometry, 47(7)(2014), 710–723.
Al Tahan, M., Al-Kaseasbeh, S., Davvaz, B., Neutrosophic quadruple hv-modules and their fundamental module, Neutrosophic Sets and Systems, 72(2024), 304–325.
Archdeacon, D., Topological graph theory. A survey, Congressus Numerantium, 115(18)(1996), 5–54.
Atanassov, K.T., Circular intuitionistic fuzzy sets, Journal of Intelligent & Fuzzy Systems, 39(5)(2020), 5981–5986.
Atanassov, K.T., Atanassov, K.T., Intuitionistic Fuzzy Sets, Springer, 1999.
Axenovich, M., Ueckerdt, T., Density of range capturing hypergraphs, Journal of Computational Geometry, 7(1)(2016), 1–21.
Bahmanian, M.A., Sajna, M., Connection and separation in hypergraphs, Theory and Applications of Graphs, 2(2)(2015), 5.

Bahmanian, M.A., Sajna, M., Hypergraphs: connection and separation, arXiv preprint arXiv:1504.04274, (2015).
Berge, C., Hypergraphs: Combinatorics of Finite Sets, volume 45, Elsevier, 1984.
Bravo, J.C.M., Piedrahita, C.J.B., Bravo, M.A.M., Pilacuan-Bonete, L.M., Integrating smed and industry 4.0 to optimize processes with plithogenic n-superhypergraphs, Neutrosophic Sets and Systems, 84(2025), 328–340.
Bretto, A., Hypergraph theory. An introduction. Mathematical Engineering, Cham: Springer, 1, 2013.
Buzaglo, S., Pinchasi, R. Rote, G., Topological hypergraphs, Thirty Essays on Geometric Graph Theory, (2013), 71–81.
Cao, Y., Integrating treesoft and hypersoft paradigms into urban elderly care evaluation: A comprehensive n-superhypergraph approach, Neutrosophic Sets and Systems, 85(2025), 852–873.
Cepeda, Y.V.M., Guevara, M.A.R., Mogro, E.J.J., Tizano, R.P., Impact of irrigation water technification on seven directories of the san juan-patoa river using plithogenic n-superhypergraphs based on environmental indicators in the canton of pujili, 2021, Neutrosophic Sets and Systems, 74(2024), 46–56.
Crespo-Berti, L.A., Argüelles, J.J.I.,Zura, M.P.V., Effectiveness of dogmatic criminal law reasoning in the permanent structure of the general theory of crime and punishment through plithogenic n-superhypergraphs, Neutrosophic Sets and Systems, (2025).
Deaconu, V., Kumjian, A., Quigg, J., Group actions on topological graphs, Ergodic Theory and Dynamical Systems, 32(5)(2012), 1527–1566.
Dekker, A.H., Colbert, B.D. et al., Network robustness and graph topology, In ACSC, 4(2004), 359–368.
Diestel, R., Graph theory, Springer (print edition), Reinhard Diestel (eBooks), 2024.
Feng, Y., Han, J., Ying, S., Gao, Y., Hypergraph isomorphism computation, IEEE Transactions on Pattern Analysis and Machine Intelligence, (2024).
Feng, Y., You, H., Zhang, Z., Ji, R., Gao, Y., Hypergraph neural networks, In Proceedings of the AAAI Conference on Artificial Intelligence, 33(2019), 3558–3565.
Fujita, T., Exploration of graph classes and concepts for superhypergraphs and n-th power mathematical structures, Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond, 3(4)(2024), 512.
Fujita, T., Hypergraph and superhypergraph approaches in electronics: A hierarchical framework for modeling power-grid hypernetworks and superhypernetworks, Journal of Energy Research and Reviews, 17(6)(2025), 102–136.
Fujita, T., An introduction and reexamination of molecular hypergraph and molecular n-superhypergraph, Asian Journal of Physical and Chemical Sciences, 13(3)(2025), 1–38.
Fujita, T., Modeling hierarchical systems in graph signal processing, electric circuits, and bond graphs via hypergraphs and superhypergraphs, Journal of Engineering Research and Reports, 27(5)(2025), 542.
Fujita, T., Multi-superhypergraph neural networks: A generalization of multi-hypergraph neural networks, Neutrosophic Computing and Machine Learning, 39(2025), 328–347.
Fujita, T., A theoretical investigation of quantum n-superhypergraph states, Neutrosophic Optimization and Intelligent Systems, 6(2025), 15–25.
Fujita, T., Unifying grain boundary networks and crystal graphs: A hypergraph and superhypergraph perspective in material sciences, Asian Journal of Advanced Research and Reports, 19(5)(2025), 344–379.
Fujita, T., Ghaib , A.A., Toward a unified theory of brain hypergraphs and symptom hypernetworks in medicine and neuroscience, Advances in Research, 26(3)(2025), 522–565.
Fujita, T., Mehmood, A., Superhypergraph attention networks, Neutrosophic Computing and Machine Learning, 40(1)(2025), 10–27.
Fujita, T., Smarandache, F., A concise study of some superhypergraph classes, Neutrosophic Sets and Systems, 77(2024), 548–593.
Fujita, T., Smarandache, F., Fundamental computational problems and algorithms for superhypergraphs, Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond (Second Volume), (2024).
Fujita, T., Smarandache, F., Superhypergraph neural networks and plithogenic graph neural networks: Theoretical foundations, Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond, 5(2025), 577.
Gao, Y., Zhang, Z., Lin, H., Zhao, X., Du, S. et al., Hypergraph learning: Methods and practices, IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(5)(2020), 2548–2566.
Gross, J.L., Yellen, J., Anderson, M., Graph Theory and Its Applications, Chapman and Hall/CRC, 2018.
Hamidi, M., Smarandache, F., Taghinezhad, M., Decision Making Based on Valued Fuzzy Superhypergraphs, Infinite Study, 2023.
Hamidi, M., Taghinezhad, M., Application of Superhypergraphs-Based Domination Number in Real World. Infinite Study, 2023.
Horn, M., De Brouwer, E., Moor, M., Moreau, Y., B. Rieck et al., Topological graph neural networks, arXiv preprint arXiv:2102.07835, (2021).
Jajodia, S., Noel, S., O’berry, B., Topological analysis of network attack vulnerability, Managing Cyber Threats: Issues, Approaches, and Challenges, (2005), 247–266.
Jech, T., Set Theory: The Third Millennium Edition, Revised and Expanded, Springer, 2003.
Khalid, H.E., Gungor, G.D., Zainal, M.A.N., Neutrosophic superhyper bi-topological spaces: Original notions and new insights, Neutrosophic Sets and Systems, 51(2022), 33–45.
Koh, K.M., Dong, F.M., Tay, E.G., Introduction to graph theory – with solutions to selected problems, In Introduction to Graph Theory, (2023).
Kynˇcl, J., Simple realizability of complete abstract topological graphs in p, Discrete & Computational Geometry, 45(3)(2011), 383–399.
Liu, J., Kok, S., Prediction of bank credit ratings using heterogeneous topological graph neural networks, arXiv preprint arXiv:2506.06293, (2025).
Magoni, D., Pansiot, J.J., Analysis of the autonomous system network topology, ACM SIGCOMM Computer Communication Review, 31(3)(2001), 26–37.
Mordeson, J.N., Nair, P.S., Fuzzy graphs and fuzzy hypergraphs, Physica, 46(2012).
Nalawade, N.B., Bapat, M.S., Jakkewad, S.G., Dhanorkar, G.A., Bhosale, D.J., Structural properties of zero-divisor hypergraph and superhypergraph over Zn: Girth and helly property, Panamerican Mathematical Journal, 35(4S)(2025), 485.
Pach, J., Pinchasi, R., Sharir, M., T´oth, G., Topological graphs with no large grids, Graphs and Combinatorics, 21(3)(2005), 355–364.
Pach, J., Solymosi, J., T´oth, G., Unavoidable configurations in complete topological graphs, Discrete & Computational Geometry, 30(2003), 311–320.
Ramos, E.L.H., Ayala, L.R.A., Macas, K.A.S., Study of factors that influence a victim’s refusal to testify for sexual reasons due to external influence using plithogenic n-superhypergraphs, Operational Research Journal, 46(2)(2025), 328–337.
Roitman, J., Introduction to Modern Set Theory, volume 8. John Wiley & Sons, 1990.
Smarandache, F., A unifying field in logics: Neutrosophic logic. In Philosophy, American Research Press, (1999), 1–141.
Smarandache, F., Plithogeny, plithogenic set, logic, probability, and statistics, arXiv preprint arXiv:1808.03948, (2018).
Smarandache, F., n-superhypergraph and plithogenic n-superhypergraph, Nidus Idearum, 7(2019), 107–113.
Smarandache, F., Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n-SuperHyperGraph, and Extension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-) HyperAlgebra, Infinite Study, (2020).
Smarandache, F., Extension of hyperalgebra to superhyperalgebra and neutrosophic superhyperalgebra (revisited), In International Conference on Computers Communications and Control,Springer, (2022), 427–432.
Smarandache, F., Introduction to the n-SuperHyperGraph-the most general form of graph today, Infinite Study, (2022).
Smarandache, F., SuperHyperFunction, SuperHyperStructure, Neutrosophic SuperHyperFunction and Neutrosophic SuperHyperStructure: Current understanding and future directions, Infinite Study, (2023).
Smarandache, F., Foundation of superhyperstructure & neutrosophic superhyperstructure, Neutrosophic Sets and Systems, 63(1)(2024), 21.
Sultana, F., Gulistan, M., Ali, M., Yaqoob, N., Khan, M. et al., A study of plithogenic graphs: applications in spreading coronavirus disease (covid-19) globally, Journal of Ambient Intelligence and Humanized Computing, 14(10)(2023), 13139–13159.
Takahashi, S., Ikeda, T., Shinagawa, Y., Kunii, T.L., Ueda, M., Algorithms for extracting correct critical points and constructing topological graphs from discrete geographical elevation data, In Computer Graphics Forum, 14(1995), 181–192.
Torra, V., Hesitant fuzzy sets, International Journal of Intelligent Systems, 25(6)(2010), 529–539.
Valencia, E.M.C., Va´squez, J.P.C., Lois, F.A´ .B., Multineutrosophic analysis of the relationship between survival and business growth in the manufacturing sector of azuay province, 2020–2023, using plithogenic n-superhypergraphs, Neutrosophic Sets and Systems, 84(2025), 341–355.
Vizuete, G.X., Gallardo, C.F., Vizuete, G., Structured analysis of a generator set lubrication system using vertex articulation in plithogenic n-superhypergraphs, Neutrosophic Sets and Systems, (2025).
Wang, H., Smarandache, F., Zhang, Y., Sunderraman, R., Single valued neutrosophic sets, Infinite Study, (2010).
Wen, T., Chen, E., Chen, Y., Tensor-view topological graph neural network, In International Conference on Artificial Intelligence and Statistics, PMLR, (2024), 4330–4338.
Xu, Z., Hesitant Fuzzy Sets Theory, volume 314, Springer, 2014.
Zadeh, L.A., Fuzzy sets, Information and Control, 8(3)(1965), 338–353.
Zhu, S., Neutrosophic n-superhypernetwork: A new approach for evaluating short video communication effectiveness in media convergence, Neutrosophic Sets and Systems, 85(2025), 1004–1017

Details

Primary Language

English

Subjects

Algebra and Number Theory, Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics), Topology, Pure Mathematics (Other)

Journal Section

Research Article

Authors

Takaaki Fujita ^*
0009-0007-1509-2728
Japan

Florentin Smarandache
0000-0002-5560-5926
United States

Publication Date

December 30, 2025

Submission Date

June 28, 2025

Acceptance Date

September 18, 2025

Published in Issue

Year 2025 Volume: 17 Number: 2

DOI

https://doi.org/10.47000/tjmcs.1729279

IZ

https://izlik.org/JA28BC27DL

Cite

RIS / Bibtex

APA

Fujita, T., & Smarandache, F. (2025). Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives. Turkish Journal of Mathematics and Computer Science, 17(2), 322-337. https://doi.org/10.47000/tjmcs.1729279

AMA

1.Fujita T, Smarandache F. Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives. TJMCS. 2025;17(2):322-337. doi:10.47000/tjmcs.1729279

Chicago

Fujita, Takaaki, and Florentin Smarandache. 2025. “Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives”. Turkish Journal of Mathematics and Computer Science 17 (2): 322-37. https://doi.org/10.47000/tjmcs.1729279.

EndNote

Fujita T, Smarandache F (December 1, 2025) Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives. Turkish Journal of Mathematics and Computer Science 17 2 322–337.

IEEE

[1]T. Fujita and F. Smarandache, “Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives”, TJMCS, vol. 17, no. 2, pp. 322–337, Dec. 2025, doi: 10.47000/tjmcs.1729279.

ISNAD

Fujita, Takaaki - Smarandache, Florentin. “Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives”. Turkish Journal of Mathematics and Computer Science 17/2 (December 1, 2025): 322-337. https://doi.org/10.47000/tjmcs.1729279.

JAMA

1.Fujita T, Smarandache F. Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives. TJMCS. 2025;17:322–337.

MLA

Fujita, Takaaki, and Florentin Smarandache. “Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives”. Turkish Journal of Mathematics and Computer Science, vol. 17, no. 2, Dec. 2025, pp. 322-37, doi:10.47000/tjmcs.1729279.

Vancouver

1.Takaaki Fujita, Florentin Smarandache. Topological Generalizations of Graphs: Integrating Hypergraph and Superhypergraph Perspectives. TJMCS. 2025 Dec. 1;17(2):322-37. doi:10.47000/tjmcs.1729279