Advances in Commutative Algebra pp 245-263 | Cite as

# Isomorphisms and Planarity of Zero-Divisor Graphs

## Abstract

Let *R* be a commutative ring with nonzero identity and *I* a proper ideal of *R*. The *zero-divisor graph* of *R*, denoted by \(\varGamma (R)\), is the graph on vertices \(R^*=R\setminus \{0\}\) where distinct vertices *x* and *y* are adjacent if and only if \(xy=0\). The *ideal-based zero-divisor graph* of *R* with respect to the ideal *I*, denoted by \(\varGamma _I(R)\), is the graph on vertices \(\{x \in R\setminus I \mid xy\in I\) for some \(y\in R\setminus I \}\), where distinct vertices *x* and *y* are adjacent if and only if \(xy\in I\). In this paper, we cover two main topics: isomorphisms and planarity of zero-divisor graphs. For each topic, we begin with a brief overview on past research on zero-divisor graphs. Whereafter, we provide extensions of that material to ideal-based zero-divisor graphs.

## Notes

### Acknowledgements

I would like to thank my advisor, David F. Anderson, for his contribution and comments in my graduate research. This material is derived from dissertation research performed at the University of Tennessee, Knoxville [18]. Some of my fondest memories are discussing the rings, graphs, and isomorphisms found in this paper with this amazing man. I affectionately remember being called a *trouble maker* when meeting with him about the error that gave birth to my work on graph isomorphisms. We spent many hours talking about many things (sometimes things other than mathematics). David, I offer you my sincerest thanks. I would also like to thank my colleagues and mentors at Maryville College for encouraging me in the transition to a faculty member. Additional thanks to Chase Worley for helping me with edits to this document.

## References

- 1.S. Akbari, H.R. Maimani, S. Yassemi, When a zero-divisor graph is planar or a complete \(r\)-partite graph. J. Algebra
**270**(1), 169–180 (2003)MathSciNetCrossRefGoogle Scholar - 2.D.F. Anderson, A. Frazier, A. Lauve, P.S. Livingston, The zero-divisor graph of a commutative ring. II, in
*Ideal Theoretic Methods in Commutative Algebra (Columbia, MO, 1999)*, Lecture Notes in Pure and Applied Mathematics, vol. 220 (Dekker, New York, 2001), pp. 61–72Google Scholar - 3.D.F. Anderson, R. Levy, J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras. J. Pure Appl. Algebra
**180**(3), 221–241 (2003)MathSciNetCrossRefGoogle Scholar - 4.D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring. J. Algebra
**217**(2), 434–447 (1999)MathSciNetCrossRefGoogle Scholar - 5.D.F. Anderson, S.B. Mulay, On the diameter and girth of a zero-divisor graph. J. Pure Appl. Algebra
**210**(2), 543–550 (2007)MathSciNetCrossRefGoogle Scholar - 6.D.F. Anderson, S. Shirinkam, Some remarks on \(\Gamma _I (R)\). Commun. Algebra, To AppearGoogle Scholar
- 7.M. Axtell, J. Coykendall, J. Stickles, Zero-divisor graphs of polynomials and power series over commutative rings. Commun. Algebra
**33**(6), 2043–2050 (2005)MathSciNetCrossRefGoogle Scholar - 8.I. Beck, Coloring of commutative rings. J. Algebra
**116**(1), 208–226 (1988)MathSciNetCrossRefGoogle Scholar - 9.R. Belshoff, J. Chapman, Planar zero-divisor graphs. J. Algebra
**316**(1), 471–480 (2007)MathSciNetCrossRefGoogle Scholar - 10.G. Chartrand, L. Lesniak,
*Graphs and Digraphs*. Chapman and Hall, CRC (1996)Google Scholar - 11.F. DeMeyer, K. Schneider, Automorphisms and zero divisor graphs of commutative rings. Int. J. Commut. Rings 93–106 (2002)Google Scholar
- 12.J.D. LaGrange, Complemented zero-divisor graphs and Boolean rings. J. Algebra
**315**(2), 600–611 (2007)MathSciNetCrossRefGoogle Scholar - 13.P.S. Livingston,
*Structure in Zero-divisor Graphs of Commutative Rings*, (Thesis, University of Tennessee Knoxville, 1997)Google Scholar - 14.H.R. Maimani, M.R. Pournaki, S. Yassemi, Zero-divisor graph with respect to an ideal. Commun. Algebra
**34**(3), 923–929 (2006)MathSciNetCrossRefGoogle Scholar - 15.S.B. Mulay, Cycles and symmetries of zero-divisors. Commun. Algebra
**30**(7), 3533–3558 (2002)MathSciNetCrossRefGoogle Scholar - 16.S.P. Redmond,
*Generalizations of the Zero-divisor Graph of a Ring*. (ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)–The University of Tennessee, 2001)Google Scholar - 17.S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring. Commun. Algebra
**31**(9), 4425–4443 (2003)MathSciNetCrossRefGoogle Scholar - 18.J.G. Smith, Jr.,
*Properties of Ideal-Based Zero-Divisor Graphs of Commutative Rings*. (ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)–The University of Tennessee, 2014)Google Scholar - 19.N.O. Smith,
*Properties of the Zero-Divisor Graph of a Ring*. (Dissertation, University of Tennessee Knoxville, 2004)Google Scholar - 20.N.O. Smith, Planar zero-divisor graphs. in
*Focus on Commutative Rings Research*, (Nova Science Publishers, New York, 2006), pp. 177–186Google Scholar - 21.N.O. Smith, Infinite planar zero-divisor graphs. Commun. Algebra
**35**(1), 171–180 (2007)MathSciNetCrossRefGoogle Scholar - 22.H.-J. Wang, Zero-divisor graphs of genus one. J. Algebra
**304**(2), 666–678 (2006)MathSciNetCrossRefGoogle Scholar - 23.S.E. Wright, Lengths of paths and cycles in zero-divisor graphs and digraphs of semigroups. Commun. Algebra
**35**(6), 1987–1991 (2007)MathSciNetCrossRefGoogle Scholar