{"id":{"repo_id":"utc","oai_identifier":"oai:scholar.utc.edu:theses-1773"},"canonical_url":"https://search.dev.ndltd.org/etd/utc/oai:scholar.utc.edu:theses-1773","repository":{"repo_id":"utc","name":"University of Tennessee - Chattanooga","base_url":"https://scholar.utc.edu/do/oai/"},"display":{"title":"On a graph parameter related to vertex labelings and its application to minimum rank problems in graph theory","abstract":"This thesis regards the minimum rank and minimum positive semidefinite rank of a simple graph. A graph parameter, called the minimum labeling degree (mld), is defined in terms of the concept of a vertex labeling of a graph, and its value is calculated for a few graph classes. It is proved here that there is a conception of mld that is independent of the notion of vertex labeling. Then, for a few other graph parameters β, including the zero-forcing number, a general inequality between mld and β is shown to hold. Further, it is demonstrated here that a certain upper bound for minimum rank in terms of minimum labeling degree holds for several classes of graphs for which minimum rank is known. Later, graphs whose complements both are K_{3,2}-free and have minimum labeling degree 2 are proved to have minimum positive semidefinite rank at most 4. Finally, two more labeling-independent conceptions of mld are given.","abstract_html":"This thesis regards the minimum rank and minimum positive semidefinite rank of a simple graph. A graph parameter, called the minimum labeling degree (mld), is defined in terms of the concept of a vertex labeling of a graph, and its value is calculated for a few graph classes. It is proved here that there is a conception of mld that is independent of the notion of vertex labeling. Then, for a few other graph parameters β, including the zero-forcing number, a general inequality between mld and β is shown to hold. Further, it is demonstrated here that a certain upper bound for minimum rank in terms of minimum labeling degree holds for several classes of graphs for which minimum rank is known. Later, graphs whose complements both are K_{3,2}-free and have minimum labeling degree 2 are proved to have minimum positive semidefinite rank at most 4. Finally, two more labeling-independent conceptions of mld are given.","abstract_has_math":false,"creators":["Plaisted, Daniel"],"institution":"University of Tennessee at Chattanooga","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Barioli, Francesco","van der Merwe, Lucas; Nichols, Roger; Walters, Terry","College of Arts and Sciences"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":null,"date_issued":"","date_published":null,"updated_at":"2026-07-24T05:46:59Z","subjects":["Graph theory","Representations of graphs"],"languages":["English","eng"],"rights":[],"rights_urls":["https://rightsstatements.org/page/InC/1.0/?language=en"],"identifier_entries":[]},"links":{"outbound_url":"https://scholar.utc.edu/theses/611","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Barioli, Francesco","van der Merwe, Lucas; Nichols, Roger; Walters, Terry","College of Arts and Sciences"]},{"key":"dc:creator","label":"Author","values":["Plaisted, Daniel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2019-08-01T07:00:00Z"]},{"key":"dc:publisher","label":"Institution","values":["University of Tennessee at Chattanooga","Chattanooga (Tenn.)"]},{"key":"dc:relation","label":"Dc Relation","values":["Masters Theses and Doctoral Dissertations"]},{"key":"dc:type","label":"Dc Type","values":["Masters theses","Text"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Graph theory","Representations of graphs"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://rightsstatements.org/page/InC/1.0/?language=en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholar.utc.edu/theses/611"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Dept. of Mathematics","M. S.; A thesis submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Master of Science."]},{"key":"dc:description.abstract","label":"Abstract","values":["This thesis regards the minimum rank and minimum positive semidefinite rank of a simple graph. A graph parameter, called the minimum labeling degree (mld), is defined in terms of the concept of a vertex labeling of a graph, and its value is calculated for a few graph classes. It is proved here that there is a conception of mld that is independent of the notion of vertex labeling. Then, for a few other graph parameters β, including the zero-forcing number, a general inequality between mld and β is shown to hold. Further, it is demonstrated here that a certain upper bound for minimum rank in terms of minimum labeling degree holds for several classes of graphs for which minimum rank is known. Later, graphs whose complements both are K_{3,2}-free and have minimum labeling degree 2 are proved to have minimum positive semidefinite rank at most 4. Finally, two more labeling-independent conceptions of mld are given."]},{"key":"dc:title","label":"Title","values":["On a graph parameter related to vertex labelings and its application to minimum rank problems in graph theory"]}]}],"canonical_facts":{"dc:contributor":["Barioli, Francesco","van der Merwe, Lucas; Nichols, Roger; Walters, Terry","College of Arts and Sciences"],"dc:creator":["Plaisted, Daniel"],"dc:date":["2019-08-01T07:00:00Z"],"dc:description":["Dept. of Mathematics","M. S.; A thesis submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Master of Science."],"dc:description.abstract":["This thesis regards the minimum rank and minimum positive semidefinite rank of a simple graph. A graph parameter, called the minimum labeling degree (mld), is defined in terms of the concept of a vertex labeling of a graph, and its value is calculated for a few graph classes. It is proved here that there is a conception of mld that is independent of the notion of vertex labeling. Then, for a few other graph parameters β, including the zero-forcing number, a general inequality between mld and β is shown to hold. Further, it is demonstrated here that a certain upper bound for minimum rank in terms of minimum labeling degree holds for several classes of graphs for which minimum rank is known. Later, graphs whose complements both are K_{3,2}-free and have minimum labeling degree 2 are proved to have minimum positive semidefinite rank at most 4. Finally, two more labeling-independent conceptions of mld are given."],"dc:identifier":["https://scholar.utc.edu/theses/611"],"dc:language":["English","eng"],"dc:publisher":["University of Tennessee at Chattanooga","Chattanooga (Tenn.)"],"dc:relation":["Masters Theses and Doctoral Dissertations"],"dc:rights":["https://rightsstatements.org/page/InC/1.0/?language=en"],"dc:subject":["Graph theory","Representations of graphs"],"dc:title":["On a graph parameter related to vertex labelings and its application to minimum rank problems in graph theory"],"dc:type":["Masters theses","Text"]},"updated_at":"2026-07-24T05:46:59Z"}