Date of Award

6-2002

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Dr. John Martino

Second Advisor

Dr. Terrell Hodge

Third Advisor

Dr. Arthur White

Fourth Advisor

Dr. Joseph Buckley

Abstract

Graphs, groups, and surfaces are all subjects of study in topological graph theory, using techniques and principles from the disciplines of graph theory, algebra, and topology. A Cayley graph provides a graphical representation of a finite group and a fixed generating set for the group; a Cayley map is a two-cell imbedding into a surface of a Cayley graph such that labeled outward-directed darts occur in the same sequence at each vertex. A dart is a directed edge. In this work, we generalize Cayley maps to allow two-cell imbeddings of graphs with loops and multiple edges.

In this thesis, we describe the distribution of inverses around a vertex for a t-balanced map. We also describe the distribution of inverses for the case where the superscript function alternates between two values a+ b/2 In this case, the inverse distribution is a blend of two at-balanced distributions. The image of ¥ corresponding to the case of the alternating superscript function is also determined. It is an extension of a dihedral group by a cyclic group. Furthermore, when k/2 is odd the extension is a semidirect product, and whena+ b/2= 1 mod t then Im ¥ is a wreath product Zk/2l Z2.

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Dissertation-Open Access

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