Skip to main content
SpringerLink
Log in

On the Connectivity and Equality of Some Graphs on Finite Semigroups

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

In this paper, we study various graphs, namely Colon power graph, cyclic graph, enhanced power graph, and commuting graph on a semigroup S. The purpose of this paper is twofold. First, we study the interconnection between the diameters of these graphs on semigroups having one idempotent. Consequently, the results on the connectedness and the diameter of the proper enhanced power graphs (or cyclic graph) of finite groups, viz., symmetric group and alternating group, are obtained. In the other part of this paper, for an arbitrary pair of these four graphs, we classify finite semigroups such that the graphs in this pair are equal. Our results generalize some of the corresponding results of these graphs on groups to semigroups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Aalipour, G., Akbari, S., Cameron, P.J., Nikandish, R., Shaveisi, F.: On the structure of the power graph and the enhanced power graph of a group. Electron. J. Combin. 24(3):3.16, 18 (2017)

  2. Afkhami, M., Jafarzadeh, A., Khashyarmanesh, K., Mohammadikhah, S.: On cyclic graphs of finite semigroups. J. Algebra Appl. 13(07), 1450035 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Araújo, J., Kinyon, M., Konieczny, J.: Minimal paths in the commuting graphs of semigroups. Eur. J. Combin. 32(2), 178–197 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Araújo, J., Bentz, W., Konieczny, J.: The commuting graph of the symmetric inverse semigroup. Isr. J. Math. 207(1), 103–149 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bauer, T., Greenfeld, B.: Commuting graphs of boundedly generated semigroups. Eur. J. Combin. 56, 40–45 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bera, S., Bhuniya, A.K.: On enhanced power graphs of finite groups. J. Algebra Appl. 17(8), 1850146 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bera, S., Dey, H.K., Mukherjee, S.K.: On the connectivity of enhanced power graphs of finite groups. Graphs Combin. 37(2), 591–603 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bosák, J.: The graphs of semigroups. In: Theory of Graphs and its Applications, pp. 119–125. Academic Press, New York (1964)

  9. Brauer, R., Fowler, K.A.: On groups of even order. Ann. Math. 62(3), 565–583 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  10. Budden, F.: Cayley graphs for some well-known groups. Math. Gazette 69(450), 271–278 (1985)

    Article  MATH  Google Scholar 

  11. Cameron, P.J.: The power graph of a finite group, II. J. Group Theory 13(6), 779–783 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cameron, P.J., Ghosh, S.: The power graph of a finite group. Discret. Math. 311(13), 1220–1222 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chakrabarty, I., Ghosh, S., Sen, M.K.: Undirected power graphs of semigroups. Semigroup Forum 78(3), 410–426 (2009)

  14. Dalal, S.: Graphs Associated with Groups and Semigroups. PhD thesis, BITS Pilani, Pilani (2021)

  15. Dalal, S., Kumar, J.: Chromatic number of the cyclic graph of infinite semigroup. Graphs Combin. 36(1), 109–113 (2020)

  16. Dalal, S., Kumar, J., Singh, S.: On the enhanced power graph of a semigroup. To Appear, Algebra Colloquium

  17. Dalal, S., Kumar, J., Singh, S.: The cyclic graph of a semigroup. arXiv:2107.11021 (2021)

  18. Doostabadi, A., Ghouchan, M.F.D.: On the connectivity of proper power graphs of finite groups. Commun. Algebra 43(10), 4305–4319 (2015)

  19. Dupont, L.A., Mendoza, D.G., Rodríguez, M.: The enhanced quotient graph of the quotient of a finite group. arXiv:1707.01127 (2017)

  20. Dupont, L.A., Mendoza, D.G., Rodríguez, M.: The rainbow connection number of enhanced power graph. arXiv:1708.07598 (2017)

  21. Giudici, M., Parker, C.: There is no upper bound for the diameter of the commuting graph of a finite group. J. Combin. Theory Ser. A 120(7), 1600–1603 (2013)

  22. Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  23. Imani, N., Sarbazi-Azad, H., Akl, S.G., Moinzadeh, P.: Chromatic sets of power graphs and their application to resource placement in multicomputer networks. Comput. Math. Appl. 58(3), 403–413 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kelarev, A.: Graph Algebras and Automata. Marcel Dekker Inc, New York (2003)

    Book  MATH  Google Scholar 

  25. Kelarev, A., Quinn, S.: A combinatorial property and power graphs of groups. Contrib. Gen. Algebra 12(58), 3–6 (2000)

    MATH  Google Scholar 

  26. Kelarev, A., Quinn, S.: Directed graphs and combinatorial properties of semigroups. J. Algebra 251(1), 16–26 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kelarev, A., Quinn, S., Smolikova, R.: Power graphs and semigroups of matrices. Bull. Austral. Math. Soc. 63(2), 341–344 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kelarev, A.V., Quinn, S.J.: A combinatorial property and power graphs of semigroups. Comment. Math. Univ. Carolin. 45(1), 1–7 (2004)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Kumar, J., Dalal, S., Baghel, V.: On the commuting graph of semidihedral group. Bull. Malays. Math. Sci. Soc. 44, 3319–3344 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ma, X., Kelarev, A., Lin, Y., Wang, K.: A survey on enhanced power graphs of finite groups. Electron. J. Graph Theory Appl. 10(1), 89–111 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mitsch, H., Petrich, M.: Restricting idempotents in \(E\)-inversive semigroups. Acta Sci. Math. (Szeged) 67(3–4), 555–570 (2001)

    MathSciNet  MATH  Google Scholar 

  33. Panda, R.P., Dalal, S., Kumar, J.: On the enhanced power graph of a finite group. Commun. Algebra 49(4), 1697–1716 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pourghobadi, K., Jafari, S.H.: The diameter of power graphs of symmetric groups. J. Algebra Appl. 17(12), 1850234, 11 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rapinchuk, A., Segev, Y., Seitz, G.: Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. J. Am. Math. Soc. 15(4), 929–978 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Segev, Y.: On finite homomorphic images of the multiplicative group of a division algebra. Ann. Math. (2) 149(1), 219–251 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Segev, Y., Seitz, G.M.: Anisotropic groups of type an and the commuting graph of finite simple groups. Pac. J. Math. 202(1), 125–225 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Shitov, Y.: A matrix ring with commuting graph of maximal diameter. J. Combin. Theory Ser. A 141, 127–135 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Shitov, Y.: Distances on the commuting graph of the ring of real matrices. Mat. Zametki 103(5), 765–768 (2018)

    MathSciNet  MATH  Google Scholar 

  40. West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Hoboken (1996)

    MATH  Google Scholar 

  41. Woodcock, T.: The commuting graph of the symmetric group S\(_{n}\). Int. J. Contemp. Math. Sci 10(6), 287–309 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

We are grateful to the anonymous referees for careful reading and helpful comments. The second authors wishes to acknowledge the support of MATRICS Grant (MTR/2018/000779) funded by SERB.

Author information

Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani, India

    Sandeep Dalal, Jitender Kumar & Siddharth Singh

Authors
  1. Sandeep Dalal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jitender Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Siddharth Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jitender Kumar.

Ethics declarations

Conflict of interest

There is no conflict of interest regarding the publishing of this paper.

Additional information

Communicated by Rosihan M. Ali.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dalal, S., Kumar, J. & Singh, S. On the Connectivity and Equality of Some Graphs on Finite Semigroups. Bull. Malays. Math. Sci. Soc. 46, 25 (2023). https://doi.org/10.1007/s40840-022-01411-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40840-022-01411-z

Keywords

Mathematics Subject Classification

Search

Navigation