## Abstract

Let R be a ring with unity. Taloukolaei and Sahebi [2] introduced the Von Neumann regular graph GV nr+(R) of a ring, whose vertex set is R and two distinct vertices x and y are adjacent if and only if x + y is a Von Neumann regular element. In this article, we investigate some new properties of GV nr+(R) such as traversability, pancyclic, unicyclic, chordal and perfect. We also investigate the domination parameters of GV nr+(R) such as dominating set, domination number, total domination number, connected domination number and give the condition when the GV nr+(R) is an excellent graph. Finally we determine the bondage number.

## Keywords

Neumann regular ring Domination parameters. Von Neumann regular graph

## Article Details

Author Biographies

### Diamond Kharkongor, North-Eastern Hill University Shillong-22, Meghalaya, India

Department of Mathematics

### Laithun Boro, North-Eastern Hill University Shillong-22, Meghalaya, India

Department of Mathematics

### Madan Mohan Singh, North-Eastern Hill University Shillong-22, Meghalaya, India

Department of Basic Sciences and Social Sciences.

### Sanghita Dutta, North-Eastern Hill University Shillong-22, Meghalaya, India

Department of Mathematics
How to Cite
Kharkongor, D., Boro, L., Singh, M. M., & Dutta, S. (2022). Some Properties of Von Neumann Regular Graphs of Rings. Journal of the Indonesian Mathematical Society, 28(2), 185–193. https://doi.org/10.22342/jims.28.2.1112.185-193

## References

1. Anderson D. F. and Badawi A., “Von Neumann regular and related elements in commutative rings”, Algebra Colloquium 19(Spec 1), (2012), 1017–1040.
2. Taloukolaei, A.J. and Sahebi, S., “Von Neumann regular graphs associated with rings”, Discrete Mathematics Algorithms and Applications, 10(03), 1850029, (2018), 1–13.
3. Ashrafi N., Maimani H. R., Pournaki M. R. and Yassemi S., “Unit Graphs Associated With Rings”, Comm. in Algebra, 38, (2010), 2851-2871.
4. Kiani S., Maimani H., Pournaki M. R. and Yassemi S., “Classification of rings with unit graphs having domination number less than four”, Rendiconti del Seminario Matematico della Universita` di Padova 133, (2015), 173–196.
5. Atiyah M. F. and MacDonald, I. G., “Introduction to commutative algebra”, Perseus Book, Camb., Mass, (1969).
6. Chartrand G. and Zhang P., “Introduction to graph theory”, Tata McGraw-hill, (2006).
7. Mekiˇs G., “Lower bounds for the domination number and the total domination number of direct product graphs”, Discrete Mathematics, 310 (23), (2010), 3310–3317.
8. J.A. Bondy, “Pancyclic graphs”, J. Combin. Theory, 11(Ser B): (1971), 41–46.
9. Tamizh Chelvam, T., and M. Balamurugan, “Complement of the generalized total graph of fields”, AKCE International Journal of Graphs and Combinatorics, 17 (3), (2020), 730–733.
10. Chelvam, T. Tamizh, S. Anukumar Kathirvel, and M. Balamurugan, “Domination in generalized unit and unitary Cayley graphs of finite rings”, Indian Journal of Pure and Applied Mathematics 51 (2), (2020), 533–556.
11. Haynes T. W., Hedetniemi S.T. and Slatar P.J., “Fundamental of domination in graphs”, Marcel Dekker. Inc., (1998).
12. Selvakumar K., and S. N. Meera, “On the genus of the Von Neumann regular graph of commutative ring”, Malaya Journal of Matematik S (1), (2020), 16–21.