Author
Swartz, Eric

Year
2009

Advisor
Seress, Akos

Abstract
A near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C. If m is the girth of Γ then the graph is called polygonal. Up until now, the only examples of 2-arc transitive polygonal graphs with arbitrarily large valency had girth no larger than seven, and the 2-arc transitive polygonal graph with largest girth had valency five and girth twenty-three (in fact, even with no restrictions on the automorphism group, there were no examples of polygonal graphs with odd girth greater than twenty-three). This thesis provides a construction of an infinite family of polygonal graphs of arbitrary girth m with 2-arc transitive automorphism groups, showing that there are 2-arc transitive polygonal graphs of arbitrarily large valency for each girth m. Furthermore, this thesis also provides a construction that, given a polygonal graph of valency r and girth m, produces a polygonal graph of valency r and girth 3m, and that the graphs constructed via this method will be 2-arc transitive if the original graph was 2-arc transitive. Finally, this thesis provides a construction of a new infinite family of near-polygonal graphs of valency 10 and a method for determining which graphs can have a given girth, which yields a few new examples of polygonal graphs.