# Isomorphisms and Planarity of Zero-Divisor Graphs

• Jesse Gerald SmithJr.
Chapter
Part of the Trends in Mathematics book series (TM)

## 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. 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)
2. 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. 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)
4. 4.
D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring. J. Algebra 217(2), 434–447 (1999)
5. 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)
6. 6.
D.F. Anderson, S. Shirinkam, Some remarks on $$\Gamma _I (R)$$. Commun. Algebra, To AppearGoogle Scholar
7. 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)
8. 8.
I. Beck, Coloring of commutative rings. J. Algebra 116(1), 208–226 (1988)
9. 9.
R. Belshoff, J. Chapman, Planar zero-divisor graphs. J. Algebra 316(1), 471–480 (2007)
10. 10.
G. Chartrand, L. Lesniak, Graphs and Digraphs. Chapman and Hall, CRC (1996)Google Scholar
11. 11.
F. DeMeyer, K. Schneider, Automorphisms and zero divisor graphs of commutative rings. Int. J. Commut. Rings 93–106 (2002)Google Scholar
12. 12.
J.D. LaGrange, Complemented zero-divisor graphs and Boolean rings. J. Algebra 315(2), 600–611 (2007)
13. 13.
P.S. Livingston, Structure in Zero-divisor Graphs of Commutative Rings, (Thesis, University of Tennessee Knoxville, 1997)Google Scholar
14. 14.
H.R. Maimani, M.R. Pournaki, S. Yassemi, Zero-divisor graph with respect to an ideal. Commun. Algebra 34(3), 923–929 (2006)
15. 15.
S.B. Mulay, Cycles and symmetries of zero-divisors. Commun. Algebra 30(7), 3533–3558 (2002)
16. 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. 17.
S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring. Commun. Algebra 31(9), 4425–4443 (2003)
18. 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. 19.
N.O. Smith, Properties of the Zero-Divisor Graph of a Ring. (Dissertation, University of Tennessee Knoxville, 2004)Google Scholar
20. 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. 21.
N.O. Smith, Infinite planar zero-divisor graphs. Commun. Algebra 35(1), 171–180 (2007)
22. 22.
H.-J. Wang, Zero-divisor graphs of genus one. J. Algebra 304(2), 666–678 (2006)
23. 23.
S.E. Wright, Lengths of paths and cycles in zero-divisor graphs and digraphs of semigroups. Commun. Algebra 35(6), 1987–1991 (2007)

© Springer Nature Singapore Pte Ltd. 2019

## Authors and Affiliations

1. 1.Maryville CollegeMaryvilleUSA