Non-Trivial Subring Perfect Codes in Unit Graph of Boolean Rings

Authors

  • Mohammad Hassan Mudaber Universiti Teknologi Malaysia
  • Nor Haniza Sarmin Universiti Teknologi Malaysia
  • Ibrahim Gambo Universiti Teknologi Malaysia

DOI:

https://doi.org/10.11113/mjfas.v18n3.2503

Keywords:

subring, Boolean ring, unit graph, perfect code

Abstract

The aim of this paper is to investigate the non-trivial subring perfect codes in a unit graph associated with the Boolean rings. We prove a subring perfect code of size , where , in the unit graphs associated with the finite Boolean rings . Moreover, we give a necessary and sufficient condition for a subring of an infinite Boolean ring  to be a perfect code of size infinity in the unit graph.

References

P. R. Halmos, “Lectures on Boolean algebras,” New York: Courier Dover Publications, 2018.

L. Gilbert, “Elements of modern algebra,” USA: Cengage Learning, 2014.

I. Beck, “ Coloring of commutative rings,” Journal of Algebra, vol. 116, no. 1, pp. 208-226, 1988.

D. F. Anderson and P. S. Livingston, “The zero-divisor graph of a commutative ring,” Journal of Algebra, vol. 217, no. 2, pp. 434-447, 1999.

D. F. Anderson and S. B. Mulay, “ On the diameter and girth of a zero-divisor graph,” Journal of Pure and Applied Algebra, vol. 210, no. 2 , pp. 543-550, 2007.

S. Bhavanari, S. Kuncham and N. Dasari, “ Prime graph of a ring,” Journal of Combinatorics, Information and System Sciences, vol. 35, pp. 27-41, 2010.

K. Patra and S. Kalita, “Prime graph of the commutative ring ,” MATEMATIKA: Malaysian Journal of Industrial and Applied Mathematics, vol. 30, pp. 59-67, 2014.

K. Pawar and S. Joshi, “Study of prime graph of a ring,” Thai Journal of Mathematics, vol. 17, no. 2, pp. 369-377, 2019.

S. S. Joshi and K. F. Pawar, “On prime graph of a ring,” International Journal of Mathematical Combinatorics, vol. 3, pp.11-22, 2019.

N. Ashrafi, H. Maimani, M. Pournaki and S. Yassemi, “ Unit graphs associated with rings,” Communication in Algebra, vol. 38, no. 8, pp. 2851-2871, 2010.

H. R. Maimani, M. Pournaki and S .Yassemi, “Necessary and sufficient conditions for unit graphs to be Hamiltonian,” Pacific Journal of Mathematics, vol. 249, no. 2, pp. 419-429, 2011.

S. Akbari, E. Estaji and M. Khorsandi, “On the unit graph of a non-commutative ring,” Algebra Colloquium, vol. 22, no. spec01, pp.817-822, 2015.

H. Su and Y. Zhou, “On the girth of the unit graph of a ring,”Journal of Algebra and Its Applications, vol. 13, no.02, p.1350082,, 2014.

H. Su and Y. Wei, “The diameter of unit graphs of rings,” Taiwanse Journal of Mathematics, vol. 23, no. 1, pp. 1-10, 2019.

H. Su, G. Tang and Y. Zhou, “Rings whose unit graphs are planar,” Publ. Math. Debrecen, vol. 86, no. 3-4, pp. 363-376, 2015.

A. Das, H. Maimani, M. Pournaki and S. Yassemi, “Nonplanarity of unit graphs and classification of the toroidal ones,” Pacific Journal of Mathematics, vol. 268, no.2, pp. 371-387, 2014.

I. J. Dejter and R. E. Giudici, “ On unitary Cayley graphs,” J. Combin. Math. Combin. Comput., vol. 18, pp. 121-124, 1995.

R. Akhtar, M. Boggess, T. Jackson-Henderson et al., “On the unitary Cayley graph of a finite ring,” Electronic Journal of Combinatorics, vol.16, pp. 1-13, 2009.

H. Su and Y. Zhou, "A characterization of rings whose unitary Cayley graphs are planar,” Journal of Algebra and Its Applications, vol. 17, no. 09, p.1850178, 2018.

J. H. Van Lint, “ A survey of perfect codes,” The Rocky Mountain Journal of Mathematics, vol. 5, no. 2, pp. 199-224, 1975.

N. Biggs, “ Perfect codes in graphs,” Journal of Combinatorial Theory Series B, vol. 15, no. 3, pp. 289-296, 1973.

H. Huang, B. Xia and Zhou, “Perfect codes in Cayley graphs,” SIAM Journal of Discrete Mathematics, vol. 32, no. 1, pp. 548-559, 2018.

X. Ma, R. Fu, X. Lu, M. Guo, and Z. Zhao, “Perfect codes in power graphs of finite groups,” Open Mathematics, vol. 15:, no.1, pp.1440-1449, 2017.

X. Ma, “Perfect codes in proper reduced power graphs of finite groups,” Communication in Algebra, vol. 48, no. 9, pp. 3881-3890, 2020.

R. Raja, “Total perfect codes in zero-divisor graphs,” (2018). arXiv preprint arXiv:1802.00723.

Downloads

Published

04-08-2022