SOME PROPERTIES OF SUPER-GRAPH OF (G (R))^c AND ITS LINE GRAPH

Document Type : Original Manuscript

Authors

1 Department of Applied Sciences, RK University, P.O. Box 360003, Rajkot, India.

2 Department of Mathematics, Government Polytechnic, P.O. Box 360003, Rajkot, India.

Abstract

Let R be a commutative ring with identity 1≠0. The comaximal ideal graph of R is the simple, undirected graph whose vertex set is the set of all proper ideals of the ring R not contained in Jacobson radical of R and two vertices I and J are adjacent in this graph if and only if I+J=R. In this article, we have discussed the graph G(R) whose vertex set is the set of all proper ideals of ring R and two vertices I and J are adjacent in this graph if and only if I+J≠R. In this article, we have discussed some interesting results about G(R) and its line graph.

Keywords

References

 1. G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shaiveisi, On the coloring of the annihilating-ideal graph of a commutative ring, Discrete Math., 312 (2012), 2620–2626.
 
2. S. Akbari, B. Miraftab, R. Nikandish, Co-maximal graphs of subgroups of groups, Canad. Math. Bull., 60(1) (2017), 12–25.
 
3. D.F. Anderson, R. Levy and J. Shapiro, Zero-divisor graphs, von Neumann regular rings and Boolean Algebras, J. Pure Appl. Algebra, 180 (2003), 221–241.
 
4. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434–447.
 
5. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Massachusetts, 1969.
 
6. R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, 2000.
 
7. I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208–226.
 
8. M. Behboodi and Z. Rakeei, The annihilating-ideal graphs of commutative rings I, J. Algebra Appl., 10(4), (2011), 727–739.
 
9. M. Behboodi and Z. Rakeei, The annihilating-ideal graphs of commutative rings II, J. Algebra Appl., 10(4) (2011), 741–753.
 
10. A. Gaur and A. Sharma, Maximal graph of a commutative ring, International J. Algebra, 7(12) (2013), 581–588.
 
11. M. I. Jinnah and S,C. Mathew, When is the comaximal graph split?, Comm. Algebra, 40(7) (2012), 2400–2404.
 
12. R. Levy and J. Shapiro, The zero-divisor graph of von Neumann regular rings, Comm. Algebra, 30(2) (2002), 745–750.
 
13. H. R. Maimani, M. Salimi, A. Sattari and S. Yassemi, Comaximal graph of commutative rings, J. Algebra, 319(4) (2008), 1801–1808.
 
14. B. Miraftab and R. Nikandish, Co-maximal graphs of two generator groups, J. Algebra Appl., 18(04) (2019), Article ID: 1950068.
 
15. B. Miraftab and R. Nikandish, Co-maximal ideal graphs of matrix algebras, Boletín de la Sociedad Matemática Mexicana, 24(1) (2018), 1–10.
 
16. 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.
 
17. K. Nazzal and M. Ghanem, On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo n, International Journal of Combinatorics, (2012), Article ID: 957284, 13 pp.
 
18. E. M. Nezhad and A. M. Rahimi, Dominating sets of the comaximal and ideal based zero-divisor graphs of commutative rings, Quaest. Math., 38(5) (2015), 613–629 
 
19. R. Nikandish and H. R. Maimani, Dominating sets of the annihilating-ideal graphs, Electron. Notes Discrete Math., 45 (2014), 17–22.
 
20. K. Samei, On the comaximal graph of a commutative ring, Canad. Math. Bull., 57(2) (2014), 413–423.
 
21. A. Sharma and A. Gaur, Line Graphs associated to the Maximal graph, J. Algebra Relat. Topics, 3(1) (2015), 1–11.
 
22. S. Visweswaran and J. Parejiya, Annihilating -ideal graphs with independence number at most four, Cogent Mathematics, 3(1) (2016), Article ID: 1155858.
 
23. S. Visweswaran and J. Parejiya, When is the complement of the comaximal graph of a commutative ring planar?, ISRN Algebra, (2014), Article ID: 736043, 8 pp.
 
24. S. Visweswaran and H. D. Patel, Some results on the complement of the annihilating ideal graph of a commutative ring, J. Algebra Appl., 14(7) (2015), Article ID: 1550099.
 
25. H. J. Wang, Graphs associated to Co-maximal ideals of commutative rings, J. Algebra, 320(7) (2008), 2917–2933.
 
26. M. Ye and T. Wu, Co-maximal ideal  graphs of commutative rings, J. Algebra Appl., 11(6) (2012), Article ID: 1250114. 
Journal of Algebraic Systems
Volume 11, Issue 2
January 2024
Pages 93-112

Files

Share

How to cite

Statistics