Cayley maps for certain cyclic groups with odd generators

For this thesis I plan on using the AMS format. My Thesis Advisor and I will meet regularly to discuss my thesis topic, prove conjectures, write results as we make progress, develop a program for calculation of genus values under certain constraints, and then organize the work for both oral and written presentation; A Cayley graph provides us with a discrete model for a finite group with specified generating set. It is desirable to represent such structures in their simplest form and also so that certain symmetries are emphasized. By simplest form, we mean to draw these graphs on surfaces so that their edges do not cross (except at their common vertices) and by emphasizing certain symmetries, we mean to impose a local symmetry by insisting that the rotation of generators emanating from each vertex of the given Cayley graph is identical. Such embeddings (or drawings) of Cayley graphs are called Cayley maps. In this thesis we begin the classification of Cayley maps for the cyclic group Z , where p is prime, with generating set O consisting of the odd integers; Due to the local symmetry that is specified at each vertex, it is possible to represent such a Cayley map by an index one voltage graph embedding (a pseudograph with one vertex and (p - 1)/2 edges). In this work, we determine the genera for Cayley maps that are covering embeddings of certain planar voltage graphs having simple region structures, namely, those planar voltage graphs consisting of only singleton or stacked loops with at most one region of size greater than 2. In addition to providing results in these cases, we also discuss the more general problem involving any arbitrary planar voltage graph.

Name thesis:degree_name

Master of Science (MS)

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematical Sciences

Grantor dc:publisher

University of Nevada, Las Vegas

Year

2003

Author dc:creator

• Griswold, Daniel

Contributors dc:contributor

• Michelle Schultz

Statement dc:rights

• IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/

Language dc:language

English

Identifier

https://oasis.library.unlv.edu/rtds/1549

OAI identifier oai:identifier

oai:oasis.library.unlv.edu:rtds-2548