# NON-NILPOTENT GRAPH OF COMMUTATIVE RINGS

Document Type : Original Manuscript

Authors

1 Department of Mathematics, Gauhati University, Guwahati-781014, India.

2 Department of Mathematics, Gauhati University, Guwahati-14, Assam, India.

Abstract

Let R be a commutative ring with unity. Let Nil(R) be the set of all nilpotent elements of R and Nil(R) = R \ Nil(R) be the set of all non-nilpotent elements of R. The non-nilpotent graph of R is a simple undirected graph GNN(R) with Nil(R) as vertex set and any two distinct vertices x and y are adjacent if and only if x+y ∈ Nil(R).
In this paper, we introduce and discuss the basic properties of the graph GNN(R). We also study the diameter and girth of GNN(R). Further, we determine the domination number and the bondage number of GNN(R). We establish a relation between diameter and domination number of GNN(R). We also establish a relation between girth and bondage number of GNN(R).

Keywords

#### References

1. S. Akbari, D. Kiani, F. Mohammadi and S. Moradi, The total graph and regular graph of a commutative ring, J. Pure Appl. Algebra, 213 (2009), 2224–2228.
2. D. F. Anderson and A. Badawi, The generalized total graph of a commutative ring, J. Algebra Appl., 12(5) (2013), fiArticle ID: 1250212, 18pp.
3. D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008), 2706–2719.
4. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434–447.
5. D. D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra, 159 (1993), 500–514.
6. S. E. Atani and S. Habibi, The total torsion element graph of a module over a commutative ring, An. St. Univ. Ovidius Constanta, 19(1) (2011), 23–34.
7. D. K. Basnet, A. Sharma And R. Dutta, Nilpotent Graph, Theory and Applications of Graphs, 8(1) (2021).
8. I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988), 208–226.
9. G. Chartrand and P. Zhang, Introduction to graph theory, Tata McGraw-Hill, 2006.
10. J. Goswami, Some domination properties of the total graph of a module with respect to singular submodule, Online Journal of Analytic Combinatorics, 15 (2020).
11. J. Goswami, K. K. Rajkhowa and H. K. Saikia, Total graph of a module with respect to singular submodule, Arab J. Math. Sci., 22 (2016), 242–249.
12. J. Goswami and H. K. Saikia, On the Line graph associated to the Total graph of a module, Matematika, 31(1) (2015), 7–13.
13. J. Goswami and M. Shabani, Domination in the entire nilpotent element graph of a module over a commutative ring, Proyecciones Journal of Mathematics, 40(6) (2021), 1411–1430.
14. T. W. Haynes, S. T. Hedetniemi and P. J. Slatar, Domination in graphsAdvanced topics, Marcel Dekker. Inc., 2000.
15. T. W. Haynes, S. T. Hedetniemi and P. J. Slatar, Fundamental of domination in graphs, Marcel Dekker. Inc., 1998.
16. S. Khojasteh and M. J. Nikmehr, The Weakly Nilpotent Graph of a Commutative Ring, Canad. Math. Bull., 60(2) (2017), 319–328.
17. J. Lambeck, Lectures on rings and modules, Blaisdell Publishing Company, Waltham, Toronto, London, 1966.
18. D. A. Mojdeh and A. M. Rahimi, Dominating sets of some graphs associated to commutative rings, Comm. Algebra, 40 (2012), 3389–3396.
19. C. Musili, Introduction to rings and modules, Narosa Publishing House, New Delhi, India, 1992.
20. M. J. Nikmehr and S. Khojasteh, On the nilpotent graph of a ring, Turkish J. Math., 37 (2013), 553–559.
21. D. Patwari, H. K. Saikia and J. Goswami, Some results on domination in the generalized total graph of a commutative ring, J. Algebra Relat. Topics, 10(1) (2022), 119–128.
22. A. Shariatinia and A. Tehranian, Domination number of total graph of module, Journal of Algebraic Structures and Their Applications, 2(1) (2015), 1–8.
23. M. H. Shekarriz, M. H. S. Haghighi and H. Sharif, On the total graph of a finite commutative ring, Comm. Algebra, 40 (2012), 2798–2807.
24. T. Tamizh Chelvam and T. Asir, Domination in the total graph of a commutative ring, J. Combin. Math. Combin. Comput., 87 (2013), 147–158