Various Topics on Graphical Structures Placed on Commutative Rings

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 { x ∈ R \ I | there exists y ∈ R \ I such that xy ∈ I }, 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.