Type of Document Master's Thesis Author Potanka, Karen Sue Author's Email Address kpotanka@vt.edu URN etd-32798-215247 Title Groups, Graphs, and Symmetry-Breaking Degree Master of Science Department Mathematics Advisory Committee

Advisor Name Title Brown, Ezra A. Committee Chair Ball, Joseph A. Committee Member Boisen, Monte B. Jr. Committee Member Keywords

- Petersen Graph
- Symmetry-Breaking
- Graph Theory
Date of Defense 1998-04-16 Availability unrestricted AbstractA labeling of a graph G is said to be r-distinguishing if noautomorphism of G preserves all of the vertex labels. The smallest

such number r for which there is an r-distinguishing labeling on

G is called the distinguishing number of G. The distinguishing set

of a group Gamma, D(Gamma), is the set of distinguishing numbers of

graphs G in which Aut(G) = Gamma. It is shown that D(Gamma) is

non-empty for any finite group Gamma. In particular,

D(D

_{n}) is found where D_{n}is the dihedral groupwith 2n elements. From there, the generalized Petersen graphs,

GP(n,k), are defined and the automorphism groups and distinguishing

numbers of such graphs are given.

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