On the power graph of Hamiltonian groups

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Shab.C., Islamic Azad University, Shabestar, Iran.

2 Department of Mathematics, Kharazmi University, 49 Mofateh Ave., Tehran, Iran.

Abstract

A finite Hamiltonian group is a non-abelian group such that all of its subgroups are normal. This paper studies the power graphs of such groups. By computing the degrees of the vertices of the power graph we manage to give a necessary condition that a non-abelian group of even order to be a Hamiltonian group. Moreover, by presenting an infinite class of non-Hamiltonian groups we prove that the given condition is not sufficient. This study results in two infinite classes of non-isomorphic Eulerian graphs of the same vertex set. The proper power graphs of all Hamiltonian groups have been explicitly identified, as well.

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References

[1] T. Anitha, R. Rajkumar and A. Gagarian, The complement of proper power graph of finite groups, Palestin J. Math, 7 No. 2 (2018) 579-597.
[2] G. Aalipour, S. Akbari, P. J. Cameron, R. Nikandish and F. Shaveisi, On the structure of the power graph and the enhanced power graph of a group, Electron. J. Comb., 24 No. 3 (2017) 3-16.
[3] J. Abawajy, A. V. Kelarev and M. Chowdhury, Power graphs: A survey, Electron. J. Graph Theory Appl., 1 No. 2 (2013) 125-147.
[4] S. Bera, H. K. Dey, K. L. Patra and B. K. Sahoo, On the domination number of proper power graphs of finite groups, Discrete Math., 348 No. 10 (2025) 114557.
[5] A. K. Bhuniya and S. K. Mukherjee, On the power graph of the direct product of two groups, Electron. Notes Discrete. Math., 63 (2017) 197-202.
[6] P. J. Cameron, The power graph of a finite group II, J. Group Theory, 13, (2010) 779-783.
[7] P. J. Cameron and S. Ghosh, The power graph of a finite group, Discrete Math., 311 (2011) 1220-1222.
[8] C. M. Campbell, E. F. Robertson, N. Ruškuc and R. M. Thomas, On semigroups defined by Coxeter-type presentations, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995) 1063-1075.
[9] I. Chakrabarty, S. Ghosh, M.K Sen, Undirected power graphs of semigroups, Semigr. Forum, 78 No. 3 (2009) 410-426.
[10] S. Chattopadhyay and P. Panigrahi, Power graphs of finite groups of even oeder, Internat. Conf. on Math. Modeling and Scientific Computations, (2012) 62-67.
[11] T. T. Chelvam and M. Sattanathan, Power graphs of finite abelian groups, Algebra and Discrete Math, 16 (2013) 33-41.
[12] D. L. Johnson, Presentation of Groups, Second Ed., Cambridge University Press, Cambridge, 1997.
[13] A. V. Kelarev and S. J. Quinn, A combinatorial property and power graphs of groups, Contrib. General Algebra, 12 (2000) 229-235.
[14] A. V. Kelarev, Graph Algebras and Automata, Marcel Dekker, New York, 2003.
[15] X. Ma, Forbidden Subgraphs in Intersection Power Graphs of Finite Groups, Algebra Colloq., 32 No. 1 (2025) 95-110.
[16] X. Ma, A. Doostabadi and K. Wang, Notes on the diameter of the complement of the power graph of a finite group, Ars Math. Contemp., 25 No. 1 (2025) P1.07.
[17] X. Ma, L. Li and G. Zhong, Perfect codes in proper intersection power graphs of finite groups, Appl. Algebra Eng. Commun. Comput. (AAECC), 36 No. 3 (2025) 543-555.
[18] M. Pourhasan and H. Doostie, On the power graphs of elementary abelian and extra special p-groups, Internat. J. Group Theory, 10 No. 2 (2021) 89-95.
[19] E. F. Robertson and Y. Ünlü, On semigroup presentations, Proc. Edinburgh Math. Soc, 36 (1993) 55-68.
[20] M. Tarnauceanu, Some combinatorial aspects of finite Hamiltonian groups, Bull. Iranian Math. Soc, 39 No. 5 (2013) 841-854.
Algebraic Structures and Their Applications

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Available Online from 21 July 2026

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