#### Date of Award

8-1998

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Arthur T. White

#### Second Advisor

Dr. Yousef Alavi

#### Third Advisor

Dr. John Martino

#### Fourth Advisor

Dr. Ping Zhang

#### Abstract

A central question in the area of topological graph theory is to find the genus of a given graph. In particular, the genus parameter has been studied for Cayley graphs. A Cayley graph is a representation of a group and a fixed generating set for that group. A group is said to be planar if there is a generating set which produces a planar Cayley graph. We say that a group is toroidal if there is a generating set that produces a toroidal Cayley graph and if there are no generating sets which produce a planar Cayley graph. Characterizations for the planar finite groups and the toroidal groups are known. In this dissertation we study graphs which model another algebraic structure, namely, the finite field.

A graph which models a finite field is called a generalized Cayley graph. This graph is produced by taking a generating set for the finite field which consists of the standard basis of additive generators for a single fixed multiplicative generator. We characterize those finite fields that have multiplicative generators which yield planar graphs. We also characterize the toroidal finite fields. To accomplish these tasks we first obtain a genus bound for the prim e order finite fields. To do this we find a bound for the number of *i*-sided regions for I <* i* < 4. Next, we use known genus results for certain classes of graphs to get bounds for the remaining finite fields. We also use these genus results to rule out potential planar and toroidal finite fields. To complete the characterizations we use various ad hoc methods.

In addition, we find the maximum genus for all generalized Cayley graphs for finite fields and, in the process, we show that the maximum genus is independent of the multiplicative generator for that finite field. In general, finding the genus of a finite field is not easy. However, we present some asymptotic results for the genus of certain classes of finite fields. We also prove that there is a finite number of finite fields, for each given genus.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Jones, Dawn M., "A Generalization of Cayley Graphs for Finite Fields" (1998). *Dissertations*. 1553.

https://scholarworks.wmich.edu/dissertations/1553