Various Topics on Graphical Structures Placed on Commutative Rings

8-2017

Dissertation

Degree Name

Doctor of Philosophy

Mathematics

Major Professor

David F. Anderson

Committee Members

Shashikant Mulay, Marie Jameson, Vasileios Maroulas, Michael Berry

Abstract

In this dissertation, we look at two types of graphs that can be placed on a commutative ring: the zero-divisor graph and the ideal-based zero-divisor graph. A zero-divisor graph is a graph whose vertices are the nonzero zero-divisors of a ring and two vertices are connected by an edge if and only if their product is 0. We classify, up to isomorphism, all commutative rings without identity that have a zero-divisor graph on 14 or fewer vertices.

An ideal-based zero-divisor graph is a generalization of the zero-divisor graph where for a ring R and ideal I the vertices are { xR \ I | there exists yR \ I such that xyI }, and two vertices are connected by an edge if and only if their product is in I. We consider cut-sets in the ideal-based zero-divisor graph. A cut-set is a set of vertices that when they and their incident edges are removed from the graph, separate the graph into several connected components. We will describe all cut-sets in the ideal-based zero-divisor graph for commutative rings with identity.

We also give some additional results about two other graphical structures, as well as include a classification of realizable zero-divisor graphs that have a specified girth and diameter for commutative rings with and without identity.

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