{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/36630"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/36630","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Groups, Graphs, and Symmetry-Breaking","abstract":"A labeling of a graph G is said to be r-distinguishing if no automorphism 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<sub>n</sub>) is found where D<sub>n</sub> is the dihedral group with 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.","abstract_html":"A labeling of a graph G is said to be r-distinguishing if no automorphism 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&lt;sub&gt;n&lt;/sub&gt;) is found where D&lt;sub&gt;n&lt;/sub&gt; is the dihedral group with 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.","abstract_has_math":false,"creators":["Potanka, Karen Sue"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Brown, Ezra A."],"committee_members":["Boisen, Monte B. Jr.","Ball, Joseph A."],"year":1998,"date_issued":"1998-04-16","date_published":"1998-04-16","updated_at":"2026-07-22T22:19:19Z","subjects":["Petersen Graph","Symmetry-Breaking","Graph Theory"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-32798-215247"],"render_values":[{"text":"etd-32798-215247","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/36630","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Brown, Ezra A."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Boisen, Monte B. 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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<sub>n</sub>) is found where D<sub>n</sub> is the dihedral group with 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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:title","label":"Title","values":["Groups, Graphs, and Symmetry-Breaking"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Brown, Ezra A."],"dc:contributor.committeemember":["Boisen, Monte B. Jr.","Ball, Joseph A."],"dc:contributor.department":["Mathematics"],"dc:creator":["Potanka, Karen Sue"],"dc:date.accessioned":["2014-03-14T20:51:19Z"],"dc:date.available":["2014-03-14T20:51:19Z","1998-04-28"],"dc:date.issued":["1998-04-16"],"dc:description.abstract":["A labeling of a graph G is said to be r-distinguishing if no automorphism 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<sub>n</sub>) is found where D<sub>n</sub> is the dihedral group with 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."],"dc:description.degree":["Master of Science"],"dc:identifier.other":["etd-32798-215247"],"dc:identifier.uri":["http://hdl.handle.net/10919/36630"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Petersen Graph","Symmetry-Breaking","Graph Theory"],"dc:title":["Groups, Graphs, and Symmetry-Breaking"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:19Z"}