#### Advisor

Eaton, Nancy [faculty advisor, Department of Mathematics]

#### Date

5-2008

#### Keywords

graph theory; knapsack; algorithm

#### Abstract

Graph theory is a branch of mathematics which studies graphs a col- lection of a set of edges and vertices used to sometimes model structures. My interest in graph theory began last semester in a math/computer sci- ence course entitled \Discrete Structures." One aspect which makes graph theory an appealing area to research is the amount of understanding that comes from a relatively short amount of time spent learning the subject material. The visual appeal of being able to draw a graph along practical applications that surface daily make graph theory a prime candidate for further research. Through research of the history of graph theory, reading of research papers, and manipulation of theorems and denitions, the importance of communication within the realm of mathematics is discovered, result- ing in an appreciation of what math research entails. By learning more about open graph theory research problems and seeing how many of the problems, if solved, provide a solution for many other research problems, the importance of math research, specically graph theory research, is demonstrated. After deepening knowledge about denitions of terms, solving home- work problems and proving theorems, I selected a certain subset of graph theory, namely the Knapsack combinatorial optimization problem, to try to create new techniques for obtaining approximations of solutions. Through experimentation and trial and error, conclusions are drawn about the open research problem. As a result of performing independent undergraduate research in graph theory, graduate math research is experienced in prepa- ration for graduate school, reiterating the importance of verbal and writ- ten communication of mathematical concepts and ideas while deepening my mathematical appreciation and understanding of graph theory.

*Knapsack Code*