On the coprime power graph of groups II
-
Mohammed A. Mutar
und
Adel Salim Tayyah
Abstract
The power graph of a group
Acknowledgments
The authors thank Daniela Bubboloni for helpful corrections, remarks and suggestions on a first draft of the present note, and for reporting to us the error in [9] corrected in [16]. We also thank the referees for valuable comments.
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Use of Large Language Models, AI and Machine Learning Tools: None declared.
-
Conflict of interest: The authors state no conflict of interest.
-
Research funding: None declared.
-
Data availability: Not applicable.
References
[1] D. F. Anderson and P. S. Livingston, “The zero-divisor graph of a commutative ring,” J. Algebra, vol. 217, no. 2, pp. 434–447, 1999. https://doi.org/10.1006/jabr.1998.7840.Suche in Google Scholar
[2] B. Zelinka, “Intersection graphs of finite abelian groups,” Czechoslovak Math. J., vol. 25, no. 2, pp. 171–174, 1975. https://doi.org/10.21136/cmj.1975.101307.Suche in Google Scholar
[3] F. Budden, “Cayley graphs for some well-known groups,” Math. Gaz., vol. 69, no. 450, pp. 271–278, 1985. https://doi.org/10.2307/3617571.Suche in Google Scholar
[4] A. V. Kelarev and S. J. Quinn, “A combinatorial property and power graphs of groups,” in Contributions to general algebra 12, Klagenfurt, Verlag Johannes Heyn, 2000, pp. 229–235.Suche in Google Scholar
[5] A. V. Kelarev and S. J. Quinn, “Directed graphs and combinatorial properties of semigroups,” J. Algebra, vol. 251, no. 1, pp. 16–26, 2002. https://doi.org/10.1006/jabr.2001.9128.Suche in Google Scholar
[6] A. V. Kelarev, S. J. Quinn, and R. Smolíková, “Power graphs and semigroups of matrices,” Bull. Aust. Math. Soc., vol. 63, no. 2, pp. 341–344, 2001. https://doi.org/10.1017/s0004972700019390.Suche in Google Scholar
[7] I. Chakrabarty, S. Ghosh, and M. K. Sen, “Undirected power graphs of semigroups,” Semigroup Forum, vol. 78, pp. 410–426, 2009, https://doi.org/10.1007/s00233-008-9132-y.Suche in Google Scholar
[8] P. J. Cameron and S. Ghosh, “The power graph of a finite group,” Discrete Math., vol. 311, no. 13, pp. 1220–1222, 2011. https://doi.org/10.1016/j.disc.2010.02.011.Suche in Google Scholar
[9] P. J. Cameron, “The power graph of a finite group, II,” J. Group Theory, vol. 13, no. 6, pp. 779–783, 2010. https://doi.org/10.1515/jgt.2010.023.Suche in Google Scholar
[10] A. Doostabadi, A. Erfanian, and M. F. D. Ghouchan, “On power graphs of finite groups with forbidden induced subgraphs,” Indag. Math., vol. 25, no. 3, pp. 525–533, 2014. https://doi.org/10.1016/j.indag.2014.01.003.Suche in Google Scholar
[11] A. Doostabadi and M. F. D. Ghouchan, “On the connectivity of proper power graphs of finite groups,” Commun. Algebra, vol. 43, no. 10, pp. 4305–4319, 2015. https://doi.org/10.1080/00927872.2014.945093.Suche in Google Scholar
[12] G. R. Pourgholi, H. Yousefi-Azari, and A. R. Ashrafi, “On the power graphs of groups with certain forbidden subgraphs,” Bull. Malays. Math. Sci. Soc., vol. 38, no. 4, pp. 1517–1525, 2015. https://doi.org/10.1007/s40840-015-0114-4.Suche in Google Scholar
[13] H. Mukherjee, “Hamiltonian cycles of power graph of abelian groups,” Afrika Mat., vol. 30, no. 7, pp. 1025–1040, 2019. https://doi.org/10.1007/s13370-019-00699-8.Suche in Google Scholar
[14] X. Ma, M. Feng, and K. Wang, “The rainbow connection number of the power graph of a finite group,” Graphs Combin., vol. 32, no. 4, pp. 1495–1504, 2016. https://doi.org/10.1007/s00373-015-1665-8.Suche in Google Scholar
[15] Y. Shitov, “Coloring the power graph of a semigroup,” Graphs Combin., vol. 33, no. 2, pp. 485–487, 2017. https://doi.org/10.1007/s00373-017-1773-8.Suche in Google Scholar
[16] D. Bubboloni and N. Pinzauti, “Critical classes of power graphs and reconstruction of directed power graphs,” J. Group Theory, vol. 28, no. 3, pp. 713–739, 2025. https://doi.org/10.1515/jgth-2023-0181.Suche in Google Scholar
[17] H. Y. Althoby, M. A. Mutar, and D. E. Otera, “A vertex separator problem for power graphs of groups,” Mathematics, vol. 13, no. 18, p. 2970, 2025. https://doi.org/10.3390/math13182970.Suche in Google Scholar
[18] M. Afkhami, A. Jafarzadeh, K. Khashyarmanesh, and S. Mohammadikhah, “On cyclic graphs of finite semigroups,” J. Algebra Appl., vol. 13, no. 7, p. 1450035, 2014, https://doi.org/10.1142/s0219498814500352.Suche in Google Scholar
[19] 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. Combin., vol. 24, no. 3, p. P3.16, 2017, https://doi.org/10.37236/6497.Suche in Google Scholar
[20] M. A. Mutar, D. E. Otera, and A. S. Tayyah, “A note on the coprime power graph of groups,” Math. Slovaca, vol. 75, no. 4, pp. 707–712, 2025, https://doi.org/10.1515/ms-2025-0052.Suche in Google Scholar
[21] S. Bera and A. K. Bhuniya, “On enhanced power graphs of finite groups,” J. Algebra Appl., vol. 17, no. 8, p. 1850146, 2018. https://doi.org/10.1142/s0219498818501463.Suche in Google Scholar
[22] J. M. Howie, Fundamentals of Semigroup Theory, Oxford, Oxford Univ. Press, 1995.10.1093/oso/9780198511946.001.0001Suche in Google Scholar
Supplementary Material
This article contains supplementary material (https://doi.org/10.1515/ms-2025-1009).
© 2025 Walter de Gruyter GmbH, Berlin/Boston
