Hypergroups Defined on Hypergraphs and their Regular Relations
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Authors: M. AL-TAHAN AND B. DAVVAZ
DOI: 10.46793/KgJMat2203.487T
Abstract:
The notion of hypergraphs, introduced around 1960, is a generalization of that of graphs and one of the initial concerns was to extend some classical results of graph theory. In this paper, we present some connections between hypergraph theory and hypergroup theory. In this regard, we construct two hypergroupoids by defining two new hyperoperations on ℍ, the set of all hypergraphs. We prove that our defined hypergroupoids are commutative hypergroups and we define hyperrings on ℍ by using the two defined hyperoperations. Moreover, we study the fundamental group, complete parts, automorphism group and strongly regular relations of one of our hypergroups.
Keywords:
Hypergraph, hypergroup, fundamental relation.
References:
[1] M. Al-Tahan and B. Davvaz, On a special single-power cyclic hypergroup and its automorphisms, Discrete Mathematics, Algorithms and Applications 7(4) (2016), 12 pages.
[2] C. Berge, Graphcs et Hypcrgraphcs, Dunod, Pum, 1972.
[3] C. Berge, Hypergraphs-Combinatorics of Finite Sets (translated from the French), North-Holland Mathematical Library, 45, North-Holland Publishing Co., Amsterdam, 1989.
[4] C. Berge, Hypergraphs generalizing bipartite graphs, in: J. Abadie (Ed.), Integer and Nonlinear Programming, North-Holland, Amsterdam, 1970, 507–509.
[5] C. Berge, The multicolorings of graphs and hypergraphs, in: Y. Alavi, D. R. Lick (Eds.), Theory and Applications of Graphs, Lecture Notes in Mathematics 642, Springer, Berlin, Heidelberg, 2978.
[6] P. Corsini, Prolegomena of Hypergroup Theory, Second Edition, Aviani Editore, Italy, 1993.
[7] P. Corsini, Hypergraphs and hypergroups, Algebra Universalis 35 (1996), 548–555.
[8] P. Corsini and V. Leoreanu, Applications of Hyperstructures Theory, Advances in Mathematics 5, Kluwer Academic Publisher, Boston, London, 2003.
[9] B. Davvaz, Polygroup Theory and Related Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, New Jersey, 2013.
[10] B. Davvaz, Semihypergroup Theory, Elsevier, London, 2016.
[11] M. De Salvo and D. Freni, Cyclic semihypergroups and hypergroups (in italian), Atti Sem. Mat. Fis. Univ. Modena 30(1) (1981), 44–59.
[12] M. Farshi, B. Davvaz and S. Mirvakili, Degree hypergroupoids associated with hypergraphs, Filomat 28(1) (2014), 119–129.
[13] M. Farshi, B. Davvaz and S. Mirvakili, Hypergraphs and hypergroups based on a special relation, Comm. Algebra 42 (2014), 3395–3406.
[14] A. Iranmanesh and M. N. Iradmusa, The combinatorial and algebraic structure of the hypergroup associated to a hypergraph, J. Mult.-Valued Logic Soft Comput. 11 (2005), 127–136.
[15] V. Leoreanu, About the simplifiable cyclic semihypergroups, Italian Journal of Pure and Applied Mathematics 7 (2000), 69–76.
[16] F. Marty, Sur une generalization de la notion de group, Congrès des Mathématiciens Scandinaves Tenu à Stockholm, 1934, 45-49.
[17] J. Mittas, Hypergroups canoniques, Math. Balkanica 2 (1972), 165–179.
[18] T. Vougiouklis, Hyperstructures and their Representations, Hadronic Press, Palm Harbor, USA, 1994.
[19] T. Vougiouklis, Cyclicity in a special class of hypergroups, Acta Universitatis Carolinae. Mathematica et Physica 22(1) (1981), 3–6.
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