eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Research article
On connected components and perfect codes of proper order graphs of finite groups; pp. 68–76

Huani Li, Shixun Lin, Xuanlong Ma

Let G be a finite group with the identity element e. The proper order graph of G, denoted by * (G), is an undirected graph with a vertex set G \ {e}, where two distinct vertices x and y are adjacent whenever o(x) | o(y) or o(y) | o(x), where o(x) and o(y) are the orders of x and y, respectively. This paper studies the perfect codes of *(G). We characterize all connected components of a proper order graph and give a necessary and sufficient condition for a connected proper order graph. We also determine the perfect codes of the proper order graphs of a few classes of finite groups, including nilpotent groups, CP-groups, dihedral groups and generalized quaternion groups.


1. Aalipour, G., Akbari, S., Cameron, P. J., Nikandish, R. and Shaveisi, F. On the structure of the power graph and the enhanced power graph of a group. Electron. J. Combin., 2017, 24(3), P3.16.

2. Abawajy, J., Kelarev, A. and Chowdhury, M. U. Power graphs: a survey. Electron. J. Graph Theory Appl., 2013, 1(2), 125–147.

3. Asboei, A. K. and Salehi, S. S. Some results on the main supergraph of finite groups. Algebra Discret. Math., 2020, 30(2), 172–178.

4. Asboei, A. K. and Salehi, S. S. The main supergraph of finite groups. New York J. Math., 2022, 28(28), 1057–1063.

5. Bubboloni, D., Iranmanesh, M. A. and Shaker, S. M. Quotient graphs for power graphs. Rendiconti del Semin. Mat. dell Univ. di Padovo, 2017, 138, 61–89.

6. Chakrabarty, I., Ghosh, S. and Sen, M. K. Undirected power graphs of semigroups. Semigroup Forum, 2009, 78, 410–426.

7. Dejter, I. J. and Serra, O. Efficient dominating sets in Cayley graphs. Discret. Appl. Math., 2003, 129(2–3), 319–328.

8. Delgado, A. L. and Wu, Y.-F. On locally finite groups in which every element has prime power order. Illinois J. Math., 2002, 46(3), 885–891.

9. Hamzeh, A. and Ashrafi, A. R. Automorphism groups of supergraphs of the power graph of a finite group. Eur. J. Combin., 2017, 60, 82–88.

10. Hamzeh, A. and Ashrafi, A. R. The order supergraph of the power graph of a finite group. Turkish J. Math., 2018, 42(4), 1978–1989.

11. Hamzeh, A. and Ashrafi, A. R. Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups. Filomat, 2017, 31(16), 5323–5334.

12. Hamzeh, A. and Ashrafi, A. R. Some remarks on the order supergraph of the power graph of a finite group. Int. Electron. J. Algebra, 2019, 26, 1–12.

13. Heden, O. A survey of perfect codes. Adv. Math. Commun., 2008, 2(2), 223–247.

14. Huang, H., Xia, B. and Zhou, S. Perfect codes in Cayley graphs. SIAM J. Discret. Math., 2018, 32(1), 548–559.

15. Johnson, D. L. Topics in the Theory of Group Presentations. Cambridge University Press, New York, 1980.

16. Kelarev, A. Graph Algebras and Automata. Marcel Dekker, New York, 2003.

17. Kelarev, A. and Quinn, S. J. A combinatorial property and power graphs of groups. Contrib. General Algebra, 2000, 12(58), 229–235.

18. Kelarev, A., Ryan, J. and Yearwood, J. Cayley graphs as classifiers for data mining: the influence of asymmetries. Discret. Math., 2009, 309(17), 5360–5369.

19. Kratochvíl, J. Perfect codes over graphs. J. Comb. Theory Ser. B, 1986, 40(2), 224–228.

20. Kumar, A., Selvaganesh, L., Cameron, P. J. and Chelvam, T. T. Recent developments on the power graph of finite groups – a survey. AKCE Int. J. Graphs Comb., 2021, 18(2), 65–94.

21. Lee, J. Independent perfect domination sets in Cayley graphs. J. Graph Theory, 2001, 37(4), 213–219.

22. Ma, X. and Su, H. On the order supergraph of the power graph of a finite group. Ric. di Mat., 2022, 71(2), 381–390.

23. Mollard, M. On perfect codes in Cartesian products of graphs. Eur. J. Comb., 2011, 32(3), 398–403.

24. Rather, B. A., Pirzada, S. and Zhou, G. F. On distance Laplacian spectra of certain finite groups. Acta Math. Sin. Engl. Ser., 2023, 39(4), 603–617.

Back to Issue