Planarity of a unit graph part -III $ |Max(R)| \geq 3 $ case (original) (raw)

Abstract

The rings considered in this article are commutative with identity 1 = 0. Recall that the unit graph of a ring R is a simple undirected graph whose vertex set is the set of all elements of the ring R and two distinct vertices x, y are adjacent in this graph if and only if x + y ∈ U(R) where U(R) is the set of all unit elements of ring R. We denote this graph by UG(R). In this article we classified rings R with |Max(R)| ≥ 3 such that UG(R) is planar.

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References (16)

  1. S. Akbari, B. Miraftar and R. Nikandish, A note on co- maximal ideal graph of commutative rings, arxiv, 2013.
  2. N. Ashrafi, H.R. Mainmani, M.R. Pournaki and S. Yassemi, Unit graph associated with rings, Comm. Al- gebra, 38(2010), 2851-2871.
  3. M.F. Atiyah and I.G. Macdonald, Introduction to Com- mutative Algebra, Addison-Wesley publishing Company, 1969.
  4. R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Springer-Verlag, New York, 2000.
  5. N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall of India Private Limited, New Delhi, 1994.
  6. M.I. Jinnah and S.C. Mathew, When is the comaximal graph split?, Comm. Algebra 40(7)(2012), 2400-2404.
  7. H.R. Maimani, M. Salimi, A. Sattari, and S. Yassemi, Comaximal graph of commutative rings, J. Algebra, 319(2008), 1801-1808.
  8. S.M. Moconja and Z.Z. Petrovic, On the structure of comaximal graphs of commutative rings with identity, Bull. Aust. Math. Soc., 83(2011), 11-21.
  9. J. Parejiya, S. Patat and P. Vadhel, Planarity of unit graph Planarity Part -I Local Case, Malaya Journal of Matem- atik, 8(3)(2020), 1155-1157.
  10. J. Parejiya, P. Vadhel and S. Patat, Planarity of unit graph Planarity Part -I Local Case, Malaya Journal of Matem- atik , 8(3)(2020), 1162-1170.
  11. K. Samei, On the comaximal graph of a commutative ring, Canad. Math. Bull., 57(2)(2014), 413-423.
  12. P.K. Sharma and S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176(1995), 124-127.
  13. S. Visweswaran and Jaydeep Parejiya, When is the com- plement of the comaximal graph of a commutative ring planar?, ISRN Algebra, 2014 (2014), 8 pages.
  14. M.Ye and T.Wu, Comaximal ideal Graphs of com- mutative rings, J. Algebra Appl., 6(2012), DOI : 10.1142/S0219498812501149.
  15. ISSN(P):2319 -3786
  16. Malaya Journal of Matematik ISSN(O):2321 -5666