**Seminars and Colloquiums**

for the week of May 8, 2017

for the week of May 8, 2017

*SPEAKERS*

**Monday**

Speaker: Darrin Weber, UTK

**Monday, May 8th**

##### PhD DISSERTATION DEFENSE

Title: Various Topics on Graphical Structures Placed on Commutative Rings

Speaker: Darrin Weber, UTK

Time: 11:00a - 12:00p

Room: Ayres 112

In this defense, 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 isomophism, commutative rings without identity that have a zero-divisor graph on 14 or fewer vertices. We also give a classification of realizable zero-divisor graphs that have a specified girth and diameter for commutative rings with and without identity.

An ideal-based zero-divisor graph is a generalization of the zero-divisor graph. It is a graph where (for a ring R and ideal I) the vertices are the elements not in I such that there exists a y (also not in I) where their product is in I. 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.

Committee members: David F. Anderson (Chair), Shashikant Mulay, Marie Jameson, Vasileios Maroulas, and Michael Berry.

*If you are interested in giving or arranging a talk for one of our seminars or colloquiums, please review our calendar. *

*If you have questions, or a date you would like to confirm, please contact colloquium AT math DOT utk DOT edu *

**Past notices:**

3/13/17 - Spring Break

Winter Break