A graph G is a (t, r)-regular graph if every collection of t independent vertices is collectively adjacent to exactly r vertices. If a graph G is (2, r)-regular where p, s, and m are positive integers, and m ≥ 2, then when n is sufficiently large, then G is isomorphic to G = Ks+mKp, where 2(p-1)+s = r. A nested (2,r)-regular graph is constructed by replacing selected cliques with a (2,r)-regular graph and joining the vertices of the peripheral cliques. For example, in a nested 's' graph when n = s + mp, we obtain n = s1+m1p1+mp. The nested 's' graph is now of the form Gs = Ks1+m1Kp1+mKp. We examine the network properties such as the average path length, clustering coefficient, and the spectrum of these nested graphs.
","abstract_html":"<p>A graph <i>G</i> is a (<i>t</i>, <i>r</i>)-regular graph if every collection of <i>t</i> independent vertices is collectively adjacent to exactly <i>r</i> vertices. If a graph <i>G</i> is (2, <i>r</i>)-regular where <i>p</i>, <i>s</i>, and <i>m</i> are positive integers, and <i>m</i> ≥ 2, then when <i>n</i> is sufficiently large, then <i>G</i> is isomorphic to <i>G = K<sub>s</sub>+mK<sub>p</sub></i>, where 2(<i>p</i>-1)+<i>s</i> = <i>r</i>. A nested (2,<i>r</i>)-regular graph is constructed by replacing selected cliques with a (2,<i>r</i>)-regular graph and joining the vertices of the peripheral cliques. For example, in a nested '<i>s</i>' graph when <i>n = s + mp</i>, we obtain <i>n = s<sub>1</sub>+m<sub>1</sub>p<sub>1</sub>+mp</i>. The nested '<i>s</i>' graph is now of the form <i>G<sub>s</sub> = K<sub>s1</sub>+m<sub>1</sub>K<sub>p1</sub>+mK<sub>p</sub></i>. We examine the network properties such as the average path length, clustering coefficient, and the spectrum of these nested graphs.</p>","abstract_has_math":false,"creators":["Brooks, Josh Daniel"],"institution":null,"degree_name":"MS (Master of Science)","degree_level":"Thesis - unrestricted","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-08-15T07:00:00Z","date_published":"2012-08-15T07:00:00Z","updated_at":"2026-07-24T02:20:42Z","subjects":["Laplacian matrix","clustering coefficient","average path length","graph theory","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/1471","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Brooks, Josh Daniel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2012-08-15T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis - unrestricted"]},{"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":["Laplacian matrix","clustering coefficient","average path length","graph theory","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/2664/viewcontent/BrooksJ072612f.pdf","https://dc.etsu.edu/etd/1471"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["A graph G is a (t, r)-regular graph if every collection of t independent vertices is collectively adjacent to exactly r vertices. If a graph G is (2, r)-regular where p, s, and m are positive integers, and m ≥ 2, then when n is sufficiently large, then G is isomorphic to G = Ks+mKp, where 2(p-1)+s = r. A nested (2,r)-regular graph is constructed by replacing selected cliques with a (2,r)-regular graph and joining the vertices of the peripheral cliques. For example, in a nested 's' graph when n = s + mp, we obtain n = s1+m1p1+mp. The nested 's' graph is now of the form Gs = Ks1+m1Kp1+mKp. We examine the network properties such as the average path length, clustering coefficient, and the spectrum of these nested graphs.
"]},{"key":"dc:title","label":"Title","values":["Nested (2,r)-regular graphs and their network properties."]}]}],"canonical_facts":{"dc:creator":["Brooks, Josh Daniel"],"dc:date.issued":["2012-08-15T07:00:00Z"],"dc:description.abstract":["A graph G is a (t, r)-regular graph if every collection of t independent vertices is collectively adjacent to exactly r vertices. If a graph G is (2, r)-regular where p, s, and m are positive integers, and m ≥ 2, then when n is sufficiently large, then G is isomorphic to G = Ks+mKp, where 2(p-1)+s = r. A nested (2,r)-regular graph is constructed by replacing selected cliques with a (2,r)-regular graph and joining the vertices of the peripheral cliques. For example, in a nested 's' graph when n = s + mp, we obtain n = s1+m1p1+mp. The nested 's' graph is now of the form Gs = Ks1+m1Kp1+mKp. We examine the network properties such as the average path length, clustering coefficient, and the spectrum of these nested graphs.
"],"dc:identifier":["https://dc.etsu.edu/context/etd/article/2664/viewcontent/BrooksJ072612f.pdf","https://dc.etsu.edu/etd/1471"],"dc:rights":["Copyright by the authors."],"dc:subject":["Laplacian matrix","clustering coefficient","average path length","graph theory","Discrete Mathematics and Combinatorics","Mathematics","Physical Sciences and Mathematics"],"dc:title":["Nested (2,r)-regular graphs and their network properties."],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis - unrestricted"],"thesis:degree_name":["MS (Master of Science)"]},"updated_at":"2026-07-24T02:20:42Z"}