#### Title

Zero Divisor Graphs and Poset Dimension

#### Date of Award

2011

#### Document Type

Dissertation

#### Degree Name

M.S. in Mathematics

#### First Advisor

Laura Sheppardson

#### Second Advisor

William Staton

#### Third Advisor

Sandra Spiroff

#### Abstract

A graph is associated to any commutative ring R where the vertices are the non-zero zero divisors of R with two vertices adjacent if x · y = 0. The zero-divisor graph has also been studied for various algebraic stuctures such as semigroups and partially ordered sets. In this paper, we will discuss some known results on zero-divisor graphs of posets as well as the concept of compactness as it relates to zero-divisor graphs. We will dicuss equivalence class graphs defined on the elements of various algebraic structures and also the reduced graph defined on the vertices of a compact graph. After introducing and discussing some known results on poset dimension, we will show that poset decomposition can be directly related to the equivalence classes represented in a reduced graph. Using this decomposition, we can build a poset of a compact graph with any dimension in a specified interval. Thus we have a device which gives us the ability to study the dimension of a poset of a zero-divisor graph.

#### Recommended Citation

Putnam, Bette Catherine, "Zero Divisor Graphs and Poset Dimension" (2011). *Electronic Theses and Dissertations*. 237.

https://egrove.olemiss.edu/etd/237