#### Title

#### Date of Award

6-1996

#### Degree Name

Doctor of Philosophy

#### Department

Statistics

#### First Advisor

Dr. Arthur White

#### Second Advisor

Dr. Gary Chartrand

#### Third Advisor

Dr. John Fink

#### Fourth Advisor

Dr. S. F. Kapoor

#### Abstract

Determining the orientable surfaces on which a given graph can be imbedded is the central problem of topological graph theory. The natural setting for studying this problem is random topological graph theory, where a probability model is defined on the space of all labeled 2-cell imbeddings of a connected graph. The major focus of this dissertation is the study of Cayley maps in the setting of random topological graph theory.

A Cayley graph provides a representation of a finite group and a fixed generating set for the group. A Cayley map is an imbedding of a Cayley graph whose vertex rotations are identical with respect to edge labels. We define a probability model on the space of all Cayley maps for a fixed group and generating set with the uniform distribution, that is, all Cayley maps are equally likely. The genus random variable is defined and studied. The minimum and maximum values of the genus random variable are called the Cayley genus and maximum Cayley genus, respectively, and the expected value of the genus random variable is called the average Cayley genus.

We determine the values of these parameters for various groups, including the symmetric groups, dihedral groups, and certain finite abelian groups. In addition, we compute the average Cayley genus for arbitrary groups having two or three generators. It is also shown that not all values between the Cayley genus and maximum Cayley genus are attained in the range of the genus random variable. Furthermore, we investigate the likelihood that the Cayley maps have the special property of being symmetrical.

Cayley maps are lifts of index one voltage graph imbeddings. In another direction, we define a bicayley map as a Cayley graph imbedding that is the lift of an index two voltage graph imbedding and study a probability model for the space of all bicayley maps for a given group and generating set

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Schultz, Michelle, "Random Cayley Maps" (1996). *Dissertations*. 1702.

https://scholarworks.wmich.edu/dissertations/1702

## Comments

Fifth Advisor: Dr. Clifton Ealy Jr.