Note for Line and Total SuperHyperGraphs: Connecting Vertices, Edges, Edges of Edges, Edges of Edges of Edges in Hierarchical Systems
Keywords:
HyperGraph, SuperHyperGraphAbstract
Hypergraphs extend classical graphs by allowing hyperedges to connect any nonempty subset of vertices,
thereby capturing complex group-level relationships. Superhypergraphs advance this framework by introducing
recursively nested powerset layers, enabling the representation of hierarchical and self-referential links among
hyperedges. A line graph encodes the adjacencies between edges of an original graph by transforming each
edge into a vertex and connecting two vertices if their corresponding edges share a common endpoint. A total
graph incorporates both the vertices and edges of the original graph as its own vertices, with edges representing
adjacency or incidence between these entities. An iterated line graph arises from the repeated application
of the line graph construction, where each iteration takes the previous line graph as its input. Similarly, an
iterated total graph is generated by iteratively applying the total graph transformation a specified number of
times. This paper investigates the hypergraph and superhypergraph analogues of these constructions, providing
a foundation for further theoretical development.
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References
M Amin Bahmanian and Mateja Sajna. Hypergraphs: connection and separation. ˇ arXiv preprint arXiv:1504.04274, 2015.
Mohammad A Bahmanian and Mateja Sajna. Connection and separation in hypergraphs. Theory and Applications of Graphs, 2(2):5,
Yifan Feng, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao. Hypergraph neural networks. In Proceedings of the AAAI
conference on artificial intelligence, volume 33, pages 3558–3565, 2019.
Yuxin Wang, Quan Gan, Xipeng Qiu, Xuanjing Huang, and David Wipf. From hypergraph energy functions to hypergraph neural
networks. In International Conference on Machine Learning, pages 35605–35623. PMLR, 2023.
Takaaki Fujita and Florentin Smarandache. Fundamental computational problems and algorithms for superhypergraphs. HyperSoft
Set Methods in Engineering, 3:32–61, 2025.
Florentin Smarandache. 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.
Mohammad Hamidi, Florentin Smarandache, and Elham Davneshvar. Spectrum of superhypergraphs via flows. Journal of Mathematics, 2022(1):9158912, 2022.
Mohammad Hamidi, Florentin Smarandache, and Mohadeseh Taghinezhad. Decision Making Based on Valued Fuzzy Superhypergraphs. Infinite Study, 2023.
Takaaki Fujita. Exploration of graph classes and concepts for superhypergraphs and n-th power mathematical structures. 2025.
Takaaki Fujita. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Biblio Publishing, 2025.
Takaaki Fujita. Rethinking strategic perception: Foundations and advancements in hypergame theory and superhypergame theory.
Prospects for Applied Mathematics and Data Analysis, 4(2):01–14, 2024.
Florentin Smarandache. Foundation of superhyperstructure & neutrosophic superhyperstructure. Neutrosophic Sets and Systems,
(1):21, 2024.
Florentin Smarandache. SuperHyperFunction, SuperHyperStructure, Neutrosophic SuperHyperFunction and Neutrosophic SuperHyperStructure: Current understanding and future directions. Infinite Study, 2023.
Florentin Smarandache. Extension of hyperalgebra to superhyperalgebra and neutrosophic superhyperalgebra (revisited). In International Conference on Computers Communications and Control, pages 427–432. Springer, 2022.
Alain Bretto. Hypergraph theory. An introduction. Mathematical Engineering. Cham: Springer, 1, 2013.
Claude Berge. Hypergraphs: combinatorics of finite sets, volume 45. Elsevier, 1984.
Takaaki Fujita. Superhypergraph neural networks and plithogenic graph neural networks: Theoretical foundations. arXiv preprint
arXiv:2412.01176, 2024.
Takaaki Fujita. Review of some superhypergraph classes: Directed, bidirected, soft, and rough. In Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond (Second Volume).
Biblio Publishing, 2024.
Chen Yan. Properties of spectra of graphs and line graphs. Applied Mathematics-A Journal of Chinese Universities, 17(3):371–376,
Robert Kincaid and Heidi Lam. Line graph explorer: scalable display of line graphs using focus+ context. In Proceedings of the
working conference on Advanced visual interfaces, pages 404–411, 2006.
Gary Chartrand. On hamiltonian line-graphs. Transactions of the American Mathematical Society, 134(3):559–566, 1968.
Arti Sharma and Atul Gaur. Line graphs associated to the maximal graph. Journal of Algebra and Related Topics, 3(1):1–11, 2015.
Alan J Hoffman. On the line graph of the complete bipartite graph. The Annals of Mathematical Statistics, 35(2):883–885, 1964.
AJ Hoffman. On the line graph of a projective plane. Proceedings of the American Mathematical Society, 16(2):297–302, 1965.
RI Tyshkevich and Vadim E Zverovich. Line hypergraphs. Discrete Mathematics, 161(1-3):265–283, 1996.
Regina I Tyshkevich and Vadim E Zverovich. Line hypergraphs: A survey. Acta Applicandae Mathematica, 52(1-3):209–222, 1998.
V Zverovich, Yu Metelsky, and P Skums. Line graphs and hypergraphs. Methods of Graph Decompositions, page 169, 2024.
AG Levin and RI Tyshkevich. Line hypergraphs. Discrete Mathematics and Applications, 3(4):407–428, 1993.
Jean-Claude Bermond, Fahir Ergincan, and Michel Syska. Line directed hypergraphs. In Cryptography and Security: From Theory
to Applications: Essays Dedicated to Jean-Jacques Quisquater on the Occasion of His 65th Birthday, pages 25–34. Springer, 2012.
Mehdi Behzad. A criterion for the planarity of the total graph of a graph. In Mathematical proceedings of the Cambridge philosophical
society, volume 63, pages 679–681. Cambridge University Press, 1967.
Yinkui Li, Ruijuan Gu, and Hui Lei. The generalized connectivity of the line graph and the total graph for the complete bipartite
graph. Applied Mathematics and Computation, 347:645–652, 2019.
DVS Sastry and B Syam Prasad Raju. Graph equations for line graphs, total graphs, middle graphs and quasi-total graphs. Discrete
Mathematics, 48(1):113–119, 1984.
Moytri Sarmah and Kuntala Patra. Line graph associated to total graph of idealization. Afrika Matematika, 27(3):485–490, 2016.
Arthur M Hobbs. Powers of graphs, line graphs, and total graphs. In Theory and Applications of Graphs: Proceedings, Michigan
May 11–15, 1976, pages 271–285. Springer, 2006.
Mukti Acharya, Atul Gaur, Amit Kumar, et al. Line signed graph of a signed total graph. Electronic Notes in Discrete Mathematics,
:389–397, 2017.
Bhavanari Satyanarayana, Devanaboina Srinivasulu, and Kuncham Syam Prasad. Line graphs and quasi-total graphs. International
Journal of Computer Applications, 105(3), 2014.
Dragos M Cvetkovic. Spectrum of the total graph of a graph. Publ. Inst. Math.(Beograd), 16(30):49–52, 1973.
Martin Knor and Ludovit Niepel. Connectivity of iterated line graphs. Discrete applied mathematics, 125(2-3):255–266, 2003.
Harishchandra S Ramane, Hanumappa B Walikar, Siddani Bhaskara Rao, B Devadas Acharya, Prabhakar Ramrao Hampiholi,
Sudhir R Jog, and Ivan Gutman. Spectra and energies of iterated line graphs of regular graphs. Applied mathematics letters,
(6):679–682, 2005.
Liming Xiong and Zhanhong Liu. Hamiltonian iterated line graphs. Discrete mathematics, 256(1-2):407–422, 2002.
Yehong Shao. A bound on connectivity of iterated line graphs. Electronic Journal of Graph Theory & Applications, 10(2), 2022.
Martin Knor and L’udov´ıt Niepel. Iterated line graphs are maximally ordered. Journal of Graph Theory, 52(2):171–180, 2006.
Harishchandra Ramane, Ivan Gutman, and Mahadevappa Gundloor. Seidel energy of iterated line graphs of regular graphs. 2015.
Gui-Xian Tian. The asymptotic behavior of (degree-) kirchhoff indices of iterated total graphs of regular graphs. Discrete Applied
Mathematics, 233:224–230, 2017.
Derek A Holton, Dingjun Lou, and KL Mcavaney. n-extendability of line graphs, power graphs, and total graphs. Australas. J
Comb., 11:215–222, 1995.
Eber Lenes, Exequiel Mallea-Zepeda, Mar´ıa Robbiano, and Jonnathan Rodr´ıguez. On the diameter and incidence energy of iterated
total graphs. Symmetry, 10(7):252, 2018.
Azriel Rosenfeld. Fuzzy graphs. In Fuzzy sets and their applications to cognitive and decision processes, pages 77–95. Elsevier,
Sankar Sahoo and Madhumangal Pal. Product of intuitionistic fuzzy graphs and degree. Journal of Intelligent & Fuzzy Systems,
(1):1059–1067, 2017.
M. G. Karunambigai, R. Parvathi, and R. Buvaneswari. Arc in intuitionistic fuzzy graphs. Notes on Intuitionistic Fuzzy Sets,
:37–47, 2011.
P. K. Kishore Kumar, S. Lavanya, Hossein Rashmanlou, and Mostafa Nouri Jouybari. Magic labeling on interval-valued intuitionistic
fuzzy graphs. J. Intell. Fuzzy Syst., 33:3999–4006, 2017.
Said Broumi, Mohamed Talea, Assia Bakali, and Florentin Smarandache. Single valued neutrosophic graphs. Journal of New theory,
(10):86–101, 2016.
Sumera Naz, Hossein Rashmanlou, and M Aslam Malik. Operations on single valued neutrosophic graphs with application. Journal
of Intelligent & Fuzzy Systems, 32(3):2137–2151, 2017.
Faruk Karaaslan and Bijan Davvaz. Properties of single-valued neutrosophic graphs. Journal of Intelligent & Fuzzy Systems,
(1):57–79, 2018.
Said Broumi, Assia Bakali, Mohamed Talea, and Florentin Smarandache. An isolated bipolar single-valued neutrosophic graphs.
In Information Systems Design and Intelligent Applications: Proceedings of Fourth International Conference INDIA 2017, pages
–822. Springer, 2018.
S Satham Hussain, Hossein Rashmonlou, R Jahir Hussain, Sankar Sahoo, Said Broumi, et al. Quadripartitioned neutrosophic graph
structures. Neutrosophic Sets and Systems, 51(1):17, 2022.
Xiaolong Shi, Saeed Kosari, Hossein Rashmanlou, Said Broumi, and S Satham Hussain. Properties of interval-valued quadripartitioned neutrosophic graphs with real-life application. Journal of Intelligent & Fuzzy Systems, 44(5):7683–7697, 2023.
Muhammad Akram and Anam Luqman. Fuzzy hypergraphs and related extensions. Springer, 2020.
Leonid S Bershtein and Alexander V Bozhenyuk. Fuzzy graphs and fuzzy hypergraphs. In Encyclopedia of Artificial Intelligence,
pages 704–709. IGI Global, 2009.
Takaaki Fujita and Florentin Smarandache. A review of the hierarchy of plithogenic, neutrosophic, and fuzzy graphs: Survey
and applications. In Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy,
Neutrosophic, Soft, Rough, and Beyond (Second Volume). Biblio Publishing, 2024.
Fazeelat Sultana, Muhammad Gulistan, Mumtaz Ali, Naveed Yaqoob, Muhammad Khan, Tabasam Rashid, and Tauseef Ahmed. A
study of plithogenic graphs: applications in spreading coronavirus disease (covid-19) globally. Journal of ambient intelligence and
humanized computing, 14(10):13139–13159, 2023.
Yixuan He, Xitong Zhang, Junjie Huang, Benedek Rozemberczki, Mihai Cucuringu, and Gesine Reinert. Pytorch geometric signed
directed: a software package on graph neural networks for signed and directed graphs. In Learning on Graphs Conference, pages
–1. PMLR, 2024.
Tahira Batool and Uzma Ahmad. A new approximation technique based on central bodies of rough fuzzy directed graphs for
agricultural development. Journal of Applied Mathematics and Computing, 70(2):1673–1705, 2024.
Enrico L. Enriquez, Grace M. Estrada, Carmelita Loquias, Reuella J. Bacalso, and Lanndon A. Ocampo. Domination in fuzzy
directed graphs. Mathematics, 2021.
Takaaki Fujita. Extensions of multidirected graphs: Fuzzy, neutrosophic, plithogenic, rough, soft, hypergraph, and superhypergraph
variants. International Journal of Topology, 2(3):11, 2025.
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