On Beck’s zero-divisor graph

Deepa Sinha and Bableen Kaur
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 4, Pages 150–157
DOI: 10.7546/nntdm.2019.25.4.150-157
Full paper (PDF, 154 Kb)

Details

Authors and affiliations

Deepa Sinha
Department of Mathematics, South Asian University
Akbar Bhawan, Chanakyapuri, New Delhi 110021, India

Bableen Kaur
Department of Mathematics, South Asian University
Akbar Bhawan, Chanakyapuri, New Delhi 110021, India

Abstract

For a commutative ring R with unity (1 ≠ 0), the zero-divisor graph of R, denoted by Γ(R), is a simple graph with vertices as elements of R and two distinct vertices are adjacent whenever the product of the vertices is zero. This article aims at gaining a deeper insight into the basic structural properties of zero-divisor graphs given by Beck.

Keywords

• Commutative ring
• Zero-divisors
• Diameter
• Girth
• Path graph
• Complete graph
• Complete bipartite graph
• Star graph

2010 Mathematics Subject Classification

• 05C25
• 05C75

References

1. Anderson, F. F., & Naseer, M. (1993). Beck’s coloring of a commutative ring, J. Algebra, 159, 500–514.
2. Anderson, F. F., Frazier, A., Lauve, A., & Livingston, P. S. (1999). The zero-divisor graph of a commutative ring, II, Lect. Notes Pure Appl. Math., 220, 61–72.
3. Anderson, D. F., & Livingston, P. S. (1999) The zero-divisor graph of a commutative ring, J. Algebra, 217, 434–447.
4. Beck, I. (1988). Coloring of commutative rings, J. Algebra, 116, 208–226.
5. Diestel, R. (2000). Graph theory, Springer-Verlag Berlin and Heidelberg GmbH & amp.
6. Dummit, D. S., & Foote, R. M. (2004). Abstract algebra, John Wiley and Sons.
7. Fine, B. (1993). Classification of finite rings of order p², Math. Mag., 248–252.
8. Ganesan, N. (1964). Properties of rings with a finite number of zero divisors, Math. Ann., 157, 215–218.
9. Harary, F. (1969). Graph theory, Addison-Wesley, Reading, MA.
10. Kaplansky, I. (1974). Commutative rings, University of Chicago Press, Chicago.
11. Levy, R., & Shapiro, J. (2002). The zero-divisor graph of Von Neumann regular rings, Comm. Algebra, 30, 745–750.
12. McDonald, B. R. (1974). Finite rings with identity, Marcel Dekker Incorporated.
13. Mulay, S. B. (2002). Cycles and symmetries of zero-divisors, Comm. Algebra, 30,
    3533–3558.
14. Redmond, S. P. (2003). An deal-based zero-divisor graph of a commutative ring, Comm. Algebra, 31, 4425–4443.