Let S ⊆ V be an arbitrary subset of vertices of a graph G = (V,E). The differential ∂(S) equals the difference between the cardinality of the set of vertices not in S but adjacent to vertices in S, and the cardinality of the set S. The differential of a graph G equals the maximum differential of any subset S of V . A set S is called a zero set if ∂(S) = 0. In this thesis we introduce the study of zero sets in graphs. We give proofs of the existence of zero sets in various kinds of graphs such as even order graphs, bipartite graphs, and graphs of maximum degree 3. We also give proofs regarding the existence of graphs which contain no zero sets and the construction of zero-free graphs from graphs which contain zero sets.
","abstract_html":"<p>Let <em>S</em> ⊆ <em>V</em> be an arbitrary subset of vertices of a graph <em>G</em> = (<em>V</em>,<em>E</em>). The differential ∂(<em>S</em>) equals the difference between the cardinality of the set of vertices not in <em>S</em> but adjacent to vertices in <em>S</em>, and the cardinality of the set <em>S</em>. The <em>differential of a graph G</em> equals the maximum differential of any subset <em>S</em> of <em>V</em> . A set <em>S</em> is called a <em>zero set</em> if ∂(<em>S</em>) = 0. In this thesis we introduce the study of zero sets in graphs. We give proofs of the existence of zero sets in various kinds of graphs such as even order graphs, bipartite graphs, and graphs of maximum degree 3. We also give proofs regarding the existence of graphs which contain no zero sets and the construction of zero-free graphs from graphs which contain zero sets.</p>","abstract_has_math":false,"creators":["Scott, Hamilton"],"institution":null,"degree_name":"MS (Master of Science)","degree_level":"Thesis - restricted","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010-05-08T07:00:00Z","date_published":"2010-05-08T07:00:00Z","updated_at":"2026-07-24T02:20:42Z","subjects":["Graph Theory","Differentials","Discrete Mathematics and Combinatorics","Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":["Copyright by the authors."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dc.etsu.edu/etd/1705","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Scott, Hamilton"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2010-05-08T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis - restricted"]},{"key":"thesis:degree_name","label":"Degree Name","values":["MS (Master of Science)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Graph Theory","Differentials","Discrete Mathematics and Combinatorics","Mathematics","Physical Sciences and Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Copyright by the authors."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dc.etsu.edu/context/etd/article/3060/viewcontent/ScottH042210f.pdf","https://dc.etsu.edu/etd/1705"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let S ⊆ V be an arbitrary subset of vertices of a graph G = (V,E). The differential ∂(S) equals the difference between the cardinality of the set of vertices not in S but adjacent to vertices in S, and the cardinality of the set S. The differential of a graph G equals the maximum differential of any subset S of V . A set S is called a zero set if ∂(S) = 0. In this thesis we introduce the study of zero sets in graphs. We give proofs of the existence of zero sets in various kinds of graphs such as even order graphs, bipartite graphs, and graphs of maximum degree 3. We also give proofs regarding the existence of graphs which contain no zero sets and the construction of zero-free graphs from graphs which contain zero sets.
"]},{"key":"dc:title","label":"Title","values":["Zero Sets in Graphs."]}]}],"canonical_facts":{"dc:creator":["Scott, Hamilton"],"dc:date.issued":["2010-05-08T07:00:00Z"],"dc:description.abstract":["Let S ⊆ V be an arbitrary subset of vertices of a graph G = (V,E). The differential ∂(S) equals the difference between the cardinality of the set of vertices not in S but adjacent to vertices in S, and the cardinality of the set S. The differential of a graph G equals the maximum differential of any subset S of V . A set S is called a zero set if ∂(S) = 0. In this thesis we introduce the study of zero sets in graphs. We give proofs of the existence of zero sets in various kinds of graphs such as even order graphs, bipartite graphs, and graphs of maximum degree 3. We also give proofs regarding the existence of graphs which contain no zero sets and the construction of zero-free graphs from graphs which contain zero sets.
"],"dc:identifier":["https://dc.etsu.edu/context/etd/article/3060/viewcontent/ScottH042210f.pdf","https://dc.etsu.edu/etd/1705"],"dc:rights":["Copyright by the authors."],"dc:subject":["Graph Theory","Differentials","Discrete Mathematics and Combinatorics","Mathematics","Physical Sciences and Mathematics"],"dc:title":["Zero Sets in Graphs."],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis - restricted"],"thesis:degree_name":["MS (Master of Science)"]},"updated_at":"2026-07-24T02:20:42Z"}