# Planar Zero-Divisor Graphs

## Date of Graduation

Summer 2006

## Degree

Master of Science in Mathematics

## Department

Mathematics

## Committee Chair

Richard Belshoff

## Abstract

Associated to every nonzero commutative ring with identity is a graph whose vertices are the nonzero zero-divisors, and such that two distinct vertices x and y are adjacent if and only if xy = 0. this graph is called the zero-divisor graph of the ring. Now assume that the ring is not a field. A result of Akbari, Maimani, and Yassemi (Journal of Algebra 270 (2003) 169-180) states that for any local ring with more than 32 elements, the zero-divisor graph is not a planar graph. In this thesis we show that for any local ring with more than 27 elements, the zero-divisor graph is not a planar graph. Moreover, we determine all finite local rings for which the zero-divisor graph is planar.

## Keywords

commutative ring, zero-divisor, zero-divisor graph, Artinian ring, local ring

## Subject Categories

Mathematics

## Copyright

© Jeremy M. Chapman

## Recommended Citation

Chapman, Jeremy M., "Planar Zero-Divisor Graphs" (2006). *MSU Graduate Theses*. 2315.

https://bearworks.missouristate.edu/theses/2315

**Dissertation/Thesis**