{"id":{"repo_id":"utc","oai_identifier":"oai:scholar.utc.edu:theses-1656"},"canonical_url":"https://search.dev.ndltd.org/etd/utc/oai:scholar.utc.edu:theses-1656","repository":{"repo_id":"utc","name":"University of Tennessee - Chattanooga","base_url":"https://scholar.utc.edu/do/oai/"},"display":{"title":"On the δ-conjecture for graphs with minimum degree |G| – 4","abstract":"For a graph G of order n, the minimum rank of G is defined to be the minimum rank among all n × n symmetric matrices whose ij-entry is nonzero precisely when {i, j} is an edge of G. The delta conjecture proposes a relationship between the minimum rank and the minimum degree of a given graph. We prove that the delta conjecture holds for several classes of graphs; in particular, we show this relationship holds for many graphs G whose minimum degree is |G| – 4. We then consider some implications of these results related to other problems involving minimum rank.","abstract_html":"For a graph G of order n, the minimum rank of G is defined to be the minimum rank among all n × n symmetric matrices whose ij-entry is nonzero precisely when {i, j} is an edge of G. The delta conjecture proposes a relationship between the minimum rank and the minimum degree of a given graph. We prove that the delta conjecture holds for several classes of graphs; in particular, we show this relationship holds for many graphs G whose minimum degree is |G| – 4. We then consider some implications of these results related to other problems involving minimum rank.","abstract_has_math":false,"creators":["Villanueva, Matthew"],"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; Ledoan, Andrew; Kuhn, Stephen","College of Arts and Sciences"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":null,"date_issued":"","date_published":null,"updated_at":"2026-07-24T05:46:51Z","subjects":["Graph theory","Mathematical analysis","Matrices","Algebras, Linear"],"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/507","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; Ledoan, Andrew; Kuhn, Stephen","College of Arts and Sciences"]},{"key":"dc:creator","label":"Author","values":["Villanueva, Matthew"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-05-01T07:00:00Z"]},{"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","Mathematical analysis","Matrices","Algebras, Linear"]}]},{"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/507"]}]},{"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":["For a graph G of order n, the minimum rank of G is defined to be the minimum rank among all n × n symmetric matrices whose ij-entry is nonzero precisely when {i, j} is an edge of G. The delta conjecture proposes a relationship between the minimum rank and the minimum degree of a given graph. We prove that the delta conjecture holds for several classes of graphs; in particular, we show this relationship holds for many graphs G whose minimum degree is |G| – 4. We then consider some implications of these results related to other problems involving minimum rank."]},{"key":"dc:title","label":"Title","values":["On the δ-conjecture for graphs with minimum degree |G| – 4"]}]}],"canonical_facts":{"dc:contributor":["Barioli, Francesco","Van der Merwe, Lucas; Ledoan, Andrew; Kuhn, Stephen","College of Arts and Sciences"],"dc:creator":["Villanueva, Matthew"],"dc:date":["2017-05-01T07:00:00Z"],"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":["For a graph G of order n, the minimum rank of G is defined to be the minimum rank among all n × n symmetric matrices whose ij-entry is nonzero precisely when {i, j} is an edge of G. The delta conjecture proposes a relationship between the minimum rank and the minimum degree of a given graph. We prove that the delta conjecture holds for several classes of graphs; in particular, we show this relationship holds for many graphs G whose minimum degree is |G| – 4. We then consider some implications of these results related to other problems involving minimum rank."],"dc:identifier":["https://scholar.utc.edu/theses/507"],"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","Mathematical analysis","Matrices","Algebras, Linear"],"dc:title":["On the δ-conjecture for graphs with minimum degree |G| – 4"],"dc:type":["Masters theses","Text"]},"updated_at":"2026-07-24T05:46:51Z"}