Revista de la
Unión Matemática Argentina
Superpower graphs of finite abelian groups
Ajay Kumar, Lavanya Selvaganesh, and T. Tamizh Chelvam

Volume 68, no. 2 (2025), pp. 761–773    

Published online (final version): October 28, 2025

https://doi.org/10.33044/revuma.4587

Download PDF

Abstract

For a finite group $G$, the superpower graph $S(G)$ is a simple undirected graph with vertex set $G$, where two distinct vertices are adjacent if and only if the order of one divides that of the other. The aim of this paper is to provide tight bounds for the vertex connectivity of $S(G)$, together with some structural properties such as maximal domination sets, Hamiltonicity, and its variations for superpower graphs of finite abelian groups. The paper concludes with some open problems.

References

  1. J. Abawajy, A. Kelarev, and M. Chowdhury, Power graphs: a survey, Electron. J. Graph Theory Appl. (EJGTA) 1 no. 2 (2013), 125–147.  DOI  MR  Zbl
  2. A. K. Asboei and S. S. Salehi, Some results on the main supergraph of finite groups, Algebra Discrete Math. 30 no. 2 (2020), 172–178.  DOI  MR  Zbl
  3. J. A. Bondy and U. S. R. Murty, Graph theory, Grad. Texts in Math. 244, Springer, New York, 2008.  MR  Zbl
  4. I. Chakrabarty, S. Ghosh, and M. K. Sen, Undirected power graphs of semigroups, Semigroup Forum 78 no. 3 (2009), 410–426.  DOI  MR  Zbl
  5. S. J. Curran and J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs—a survey, Discrete Math. 156 no. 1-3 (1996), 1–18.  DOI  MR  Zbl
  6. J. A. Gallian, Contemporary abstract algebra, fourth ed., Narosa, New Delhi, 1999.  Zbl
  7. A. Hamzeh and A. R. Ashrafi, Automorphism groups of supergraphs of the power graph of a finite group, European J. Combin. 60 (2017), 82–88.  DOI  MR  Zbl
  8. A. Hamzeh and A. R. Ashrafi, Spectrum and $L$-spectrum of the power graph and its main supergraph for certain finite groups, Filomat 31 no. 16 (2017), 5323–5334.  DOI  MR  Zbl
  9. A. Hamzeh and A. R. Ashrafi, The order supergraph of the power graph of a finite group, Turkish J. Math. 42 no. 4 (2018), 1978–1989.  DOI  MR  Zbl
  10. A. Hamzeh and A. R. Ashrafi, Some remarks on the order supergraph of the power graph of a finite group, Int. Electron. J. Algebra 26 (2019), 1–12.  DOI  MR  Zbl
  11. A. Kelarev, Ring constructions and applications, Series in Algebra 9, World Scientific, River Edge, NJ, 2002.  MR  Zbl
  12. A. Kelarev, Graph algebras and automata, Monogr. Textbooks Pure Appl. Math. 257, Marcel Dekker, New York, 2003.  MR  Zbl
  13. A. Kelarev, Labelled Cayley graphs and minimal automata, Australas. J. Combin. 30 (2004), 95–101.  MR  Zbl
  14. A. Kelarev, J. Ryan, and J. Yearwood, Cayley graphs as classifiers for data mining: the influence of asymmetries, Discrete Math. 309 no. 17 (2009), 5360–5369.  DOI  MR  Zbl
  15. A. Kumar, L. Selvaganesh, P. J. Cameron, and T. Tamizh Chelvam, Recent developments on the power graph of finite groups—a survey, AKCE Int. J. Graphs Comb. 18 no. 2 (2021), 65–94.  DOI  MR  Zbl
  16. A. Kumar, L. Selvaganesh, and T. Tamizh Chelvam, Structural properties of super power graph of dihedral group $D_{2n}$, preprint, 2021.
  17. C. H. Li, On isomorphisms of finite Cayley graphs—a survey, Discrete Math. 256 no. 1-2 (2002), 301–334.  DOI  MR  Zbl
  18. X. Ma and H. Su, On the order supergraph of the power graph of a finite group, Ric. Mat. 71 no. 2 (2022), 381–390.  DOI  MR  Zbl
  19. M. Mirzargar, A survey on the automorphism groups of the commuting graphs and power graphs, Facta Univ. Ser. Math. Inform. 34 no. 4 (2019), 729–743.  DOI  MR  Zbl
  20. D. Witte and J. A. Gallian, A survey: Hamiltonian cycles in Cayley graphs, Discrete Math. 51 no. 3 (1984), 293–304.  DOI  MR  Zbl