# Research

## Discrete Mathematics

### Monday, April 23, 2001

Title: Domination Numbers of Circulant Graphs
Speaker: Vicky Wood
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

I will define and discuss circulant graphs and some of their characteristics. Primarily, the meaning of domination number and how I used Maple to find these numbers for most graphs through $$n=2$$ will be explained. Secondarily, I will touch upon connectivity, isomorphism and one technique used to eliminate isomorphic graphs from the data for better computational efficiency. In addition, I will briefly mention 2-packing numbers and explore possible application ideas for domination numbers of circulant graphs. Finally, I will wrap up the session with a challenge to others to look for patterns in the data I generated that might lead to theorems about domination number in circulant graphs.

### Monday, April 16, 2001

Title: Generating Caley Graphs
Speaker: Daniela (Genova) Filipov
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

Cayley graphs appear in problems related to edge-coloring, planarity, and symmetries in a graph. I will define Cayley graphs, superposition of graphs, Möbius Ladder, and twisted prismatic identification. I will discuss the structure of Cayley graphs using these concepts. This material is taken from a recent paper by Abreu and Guidici. The main goal of the paper is to give graph-theoretic descriptions of the Cayley graphs for all groups of small order.

### Monday, April 9, 2001

Title: More on Linear Cellular Automata
Speaker: Edwin Clark
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

This talk will be more or less independent of the talk last week. I will discuss several definitions of linear cellular automata and give a few basic results. In particular I will give a few examples and discuss implications of Fitting's Lemma for the transition diagram of global states.

### Monday, April 2, 2001

Title: Linear Cellular Automata and Garden-of-Eden Configurations
Speaker: Edwin Clark
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

I will discuss the definition of linear cellular automata (on graphs) and prove a few general facts concerning such automata. I will also present Sutner's proof of Sutner's Theorem: The all-ones problem has a solution in any finite graph.

The proof requires some basic facts from linear algebra over a finite field which will be reviewed prior to the presentation of the proof.

### Monday, March 26, 2001

Title: Cyclic Homology of Algebras, Part IV
Time: 3:00pm‐4:00pm
Place: PHY 108

### Monday, March 5, 2001

Title: Cyclic Homology of Algebras, Part III
Time: 3:00pm‐4:00pm
Place: PHY 108

### Monday, February 19, 2001

Title: Cyclic Homology of Algebras, Part II
Time: 3:00pm‐4:00pm
Place: PHY 108

### Monday, February 12, 2001

Title: Cyclic Homology of Algebras
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

We will define Hochschild and Cyclic Homologies of associative Algebras, give some examples and the relationship between the two.

### Monday, February 5, 2001

Title: Making Bigger Quandles, Part II
Speaker: Masahiko Saito
Time: 3:00pm‐4:00pm
Place: PHY 108

I will show how the 3rd cohomology groups of quandles are related to making even bigger quandles. I will also discuss relations to knots.

### Monday, January 29, 2001

Title: Making Bigger Quandles
Speaker: Masahiko Saito
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

A quandle is a set with a self-distributive binary operation with a few other properties. I will review some basics of quandles, and discuss a construction of bigger quandles from given smaller quandles.

### Monday, January 22, 2001

Title: Organizational Meeting
Time: 3:00pm‐4:00pm
Place: PHY 108

Summary

This meeting will also be a special session. Natasha Jonoska has asked for volunteers to go to nearby colleges and make presentations on “How to Choose and Apply to a Math Graduate School”. She has even prepared a collection of overhead slides of volunteers to use. Masahico Saito will go over the slides. Anyone, discrete or otherwise, who is interested in volunteering is invited.