Skip to main content
SpringerLink
Log in

Domination in generalized unit and unitary Cayley graphs of finite rings

  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

Let R be a finite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph Γ is the simple undirected graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u in U(R) such that x + uy is a unit in R. Also, \(\overline{\Gamma}\) denotes the complement of Γ. In this paper, we find the domination number γ of Γ as well as \(\overline{\Gamma}\) and characterize all γ-sets in Γ and \(\overline{\Gamma}\). Also, we obtain the bondage number of Γ. Further, we obtain the values of some domination parameters like independent, strong and weak domination numbers of \(\overline{\Gamma}\).

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.

Similar content being viewed by others

References

  1. T. Asir and T. Tamizh Chelvam, On the genus of generalized unit and unitary Cayley graphs of a commutative ring, Acta Math. Hungar., 142(2) (2014), 444–458.

    Article  MathSciNet  Google Scholar 

  2. T. Asir and T. Tamizh Chelvam, On the genus two characterizations unit, unitary Cayley and comaximal graphs, ARS Combinatoria, 138 (2018), 77–91.

    MathSciNet  MATH  Google Scholar 

  3. J. F. Fink, M. S. Jacobson, L. F. Kinch, and J. Roberts, The bondage number of a graph, Discrete Math., 86 (1990), 47–57.

    Article  MathSciNet  Google Scholar 

  4. G. Chartrand and P. Zhang, Introduction to graph theory, Tata McGraw-Hill Edition, 2006.

    MATH  Google Scholar 

  5. T. W. Haynes, S. T. Hedetniemi and P. J. Slatar, Fundamental of domination in graphs, Marcel Dekker Inc., 1998.

    Google Scholar 

  6. K. Khashyarmanesh and M. R. Khorsandi, A generalization of the unit and unitary Cayley graphs of a commutative ring, Acta Math. Hungar., 137(4) (2012), 242–253.

    Article  MathSciNet  Google Scholar 

  7. S. Kiani, H. R. Maimani, M. R. Pournaki and S. Yassemi, Classification of rings with unit graphs having domination number less than four, Rend. Sem. Mat. Univ. Padova, 133 (2015), DOI https://doi.org/10.4171/RSMUP/133-9.

  8. A. R. Naghipour and M. Rezagholibeigi, A refinement of the unit and unitary Cayley graphs of a finite ring, Bull. Korean Math. Soc., 53(4) (2016), 1197–1211.

    Article  MathSciNet  Google Scholar 

  9. T. Tamizh Chelvam and S. Anukumar Kathirvel, Generalized unit and unitary Cayley graphs of finite rings, J. Algebra Appl., 2019, https://doi.org/10.1142/S0219498819500063.

    Google Scholar 

  10. T. Tamizh Chelvam, T. Asir and K. Selvakumar, On the domination in graphs from commutative rings, Algebra and its applications, ICAA, Aligarh, India, December 2014, S. Tariq Rizvi et al. (eds), Springer Proceedings in Mathematics and Statistics, 174, DOI: https://doi.org/10.1007/978-981-10-1651-6.

Download references

Acknowledgement

This research work is supported by the SERB Project No. SR/S4/MS.806/13 of Science and Engineering Research Board, Government of India through the first author.

Author information

Authors and Affiliations

  1. Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, 627 012, Tamil Nadu, India

    T. Tamizh Chelvam, S. Anukumar Kathirvel & M. Balamurugan

Authors
  1. T. Tamizh Chelvam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Anukumar Kathirvel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Balamurugan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to T. Tamizh Chelvam, S. Anukumar Kathirvel or M. Balamurugan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tamizh Chelvam, T., Anukumar Kathirvel, S. & Balamurugan, M. Domination in generalized unit and unitary Cayley graphs of finite rings. Indian J Pure Appl Math 51, 533–556 (2020). https://doi.org/10.1007/s13226-020-0415-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-020-0415-7

Key words

2010 Mathematics Subject Classification

Search

Navigation