Date of Award
Doctor of Philosophy
Dr. Arthur T. White
Dr. Alfred Boals
Dr. Donald Goldsmith
Dr. John Petro
This dissertation develops formulas for the number of congruence classes of maps of complete, complete bi-partite, complete tripartite, and complete n-partite graphs; these congruence classes correspond to unlabeled imbeddings. The method employed for the enumeration is an extension of that used by Mull, Rieper, and White in 1988. We let the automorphism group act on the set of rotations and use Burnside's Lemma to count orbits for these rotations. Compatible permutations are introduced to determine those automorphisms actually contributing to the number of orbits.
The complete n-partite formula is shown to generalize those of the other three families of graphs. These congruence classes are classified using the hierarchy of Kountanis, Mull, and Rashidi (1989). A new parameter is introduced to random topological graph theory: the average number of symmetries of the maps of a graph. This parameter is evaluated for arbitrary connected graphs G, in terms of the degree sequence, the number of graph automorphisms, and the number of congruence classes of maps of G. Finally, the computer program created to classify the congruence classes, and which verified the formulas for small cases, is presented.
Mull, Bruce P., "Enumerating the Orientable 2-Cell Imbeddings of Complete N-Partite Graphs" (1990). Dissertations. 2093.