{"id":{"repo_id":"wku-diss","oai_identifier":"oai:digitalcommons.wku.edu:theses-1409"},"canonical_url":"https://search.dev.ndltd.org/etd/wku-diss/oai:digitalcommons.wku.edu:theses-1409","repository":{"repo_id":"wku-diss","name":"Western Kentucky University","base_url":"https://digitalcommons.wku.edu/do/oai/"},"display":{"title":"Loop Edge Estimation in 4-Regular Hamiltonian Graphs","abstract":"In knot theory, a knot is defined as a closed, non-self-intersecting curve embedded in three-dimensional space that cannot be untangled to produce a simple planar loop. A mathematical knot is essentially a conventional knot tied with rope where the ends of the rope have been glued together. One way to sample large knots is based on choosing a 4-regular Hamiltonian planar graph. A method for generating rooted 4-regular Hamiltonian planar graphs with n vertices is discussed in this thesis. In the generation process of these graphs, some vertices are introduced that can be easily eliminated from the resulting knot diagram. The main result of this thesis is the estimation of the expected number of loop edges in a 4-regular Hamiltonian planar graphs of n vertices; in particular, it is shown that the expected number of loop edges L(n) in such a graph has asymptotic order n/6.","abstract_html":"In knot theory, a knot is defined as a closed, non-self-intersecting curve embedded in three-dimensional space that cannot be untangled to produce a simple planar loop. A mathematical knot is essentially a conventional knot tied with rope where the ends of the rope have been glued together. One way to sample large knots is based on choosing a 4-regular Hamiltonian planar graph. A method for generating rooted 4-regular Hamiltonian planar graphs with n vertices is discussed in this thesis. In the generation process of these graphs, some vertices are introduced that can be easily eliminated from the resulting knot diagram. The main result of this thesis is the estimation of the expected number of loop edges in a 4-regular Hamiltonian planar graphs of n vertices; in particular, it is shown that the expected number of loop edges L(n) in such a graph has asymptotic order n/6.","abstract_has_math":false,"creators":["Madden, Yale"],"institution":null,"degree_name":"Master of Science","degree_level":null,"degree_discipline":"Department of Mathematics and Computer Science","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2007,"date_issued":"2007-05-01T07:00:00Z","date_published":"2007-05-01T07:00:00Z","updated_at":"2026-07-24T06:07:28Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wku.edu/theses/406","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Madden, Yale"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematics and Computer Science"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wku.edu/theses/406"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In knot theory, a knot is defined as a closed, non-self-intersecting curve embedded in three-dimensional space that cannot be untangled to produce a simple planar loop. A mathematical knot is essentially a conventional knot tied with rope where the ends of the rope have been glued together. One way to sample large knots is based on choosing a 4-regular Hamiltonian planar graph. A method for generating rooted 4-regular Hamiltonian planar graphs with n vertices is discussed in this thesis. In the generation process of these graphs, some vertices are introduced that can be easily eliminated from the resulting knot diagram. The main result of this thesis is the estimation of the expected number of loop edges in a 4-regular Hamiltonian planar graphs of n vertices; in particular, it is shown that the expected number of loop edges L(n) in such a graph has asymptotic order n/6."]},{"key":"dc:title","label":"Title","values":["Loop Edge Estimation in 4-Regular Hamiltonian Graphs"]}]}],"canonical_facts":{"dc:creator":["Madden, Yale"],"dc:description.abstract":["In knot theory, a knot is defined as a closed, non-self-intersecting curve embedded in three-dimensional space that cannot be untangled to produce a simple planar loop. A mathematical knot is essentially a conventional knot tied with rope where the ends of the rope have been glued together. One way to sample large knots is based on choosing a 4-regular Hamiltonian planar graph. A method for generating rooted 4-regular Hamiltonian planar graphs with n vertices is discussed in this thesis. In the generation process of these graphs, some vertices are introduced that can be easily eliminated from the resulting knot diagram. The main result of this thesis is the estimation of the expected number of loop edges in a 4-regular Hamiltonian planar graphs of n vertices; in particular, it is shown that the expected number of loop edges L(n) in such a graph has asymptotic order n/6."],"dc:identifier":["https://digitalcommons.wku.edu/theses/406"],"dc:subject":["Mathematics"],"dc:title":["Loop Edge Estimation in 4-Regular Hamiltonian Graphs"],"dc:type":["Thesis"],"thesis:degree_discipline":["Department of Mathematics and Computer Science"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T06:07:28Z"}