{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71204"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71204","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Distance-Regular Graphs and Generalizations","abstract":"A distance-transitive graph (GAMMA) is an undirected, locally finite graph where for any vertices u,v,x,y, (PAR-DIFF)(u,v) = (PAR-DIFF)(x,y) implies (sigma)u = x and (sigma)v = y for some automorphism (sigma) of (GAMMA). Distance-transitive graphs have certain combinatorial properties, which can be studied independently; a graph with these properties is called distance-regular.","abstract_html":"A distance-transitive graph (GAMMA) is an undirected, locally finite graph where for any vertices u,v,x,y, (PAR-DIFF)(u,v) = (PAR-DIFF)(x,y) implies (sigma)u = x and (sigma)v = y for some automorphism (sigma) of (GAMMA). Distance-transitive graphs have certain combinatorial properties, which can be studied independently; a graph with these properties is called distance-regular.","abstract_has_math":false,"creators":["Terwilliger, Paul M."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:03Z","date_published":"2014-12-16T06:18:03Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8218574"],"render_values":[{"text":"(UMI)AAI8218574","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71204","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Terwilliger, Paul M."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:03Z","10000-01-01","1982"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"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":["http://hdl.handle.net/2142/71204","(UMI)AAI8218574"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A distance-transitive graph (GAMMA) is an undirected, locally finite graph where for any vertices u,v,x,y, (PAR-DIFF)(u,v) = (PAR-DIFF)(x,y) implies (sigma)u = x and (sigma)v = y for some automorphism (sigma) of (GAMMA). Distance-transitive graphs have certain combinatorial properties, which can be studied independently; a graph with these properties is called distance-regular.","We deal first with distance-regular graphs with girth 3 and 4. We show any such graph which contains a cycle (v(,1),v(,2),v(,3),v(,4),v(,1)) where (PAR-DIFF)(v(,1),v(,3)) = (PAR-DIFF)(v(,2),v(,4)) = 2 is finite with diameter d bounded by its valency k. Next we obtain new "feasibility conditions" that the intersection numbers of arbitrary distance-regular graphs with girth 3 or 4 must satisfy. We use these conditions to generate the intersection array of certain distance-regular graphs from their valency and three other parameters.","We then study distance-regular graphs with arbitrary girth and extend the ideas used in the girth 3 or 4 case to obtain a bound on the diameter of a class of distance-regular graphs, including all those with even girth. We then introduce a generalization of a distance-regular graph called a (s,c,a,k)-graph, which possesses enough of the local structure of a distance-regular graph to enable us to find bounds on their diameters in some cases. Finally we obtain lower bounds on the eigenvalue multiplicities for distance-regular graphs in terms of their valence and girth, yielding additional feasibility conditions for their intersection arrays.","Made available in DSpace on 2014-12-16T06:18:03Z (GMT). No. of bitstreams: 1 8218574.pdf: 2612218 bytes, checksum: a2c414f514218467b97cd3b109618e49 (MD5) Previous issue date: 1982","Embargo set by: Seth Robbins for item 71370 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","115 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982."]},{"key":"dc:title","label":"Title","values":["Distance-Regular Graphs and Generalizations"]}]}],"canonical_facts":{"dc:creator":["Terwilliger, Paul M."],"dc:date":["2014-12-16T06:18:03Z","10000-01-01","1982"],"dc:description":["A distance-transitive graph (GAMMA) is an undirected, locally finite graph where for any vertices u,v,x,y, (PAR-DIFF)(u,v) = (PAR-DIFF)(x,y) implies (sigma)u = x and (sigma)v = y for some automorphism (sigma) of (GAMMA). Distance-transitive graphs have certain combinatorial properties, which can be studied independently; a graph with these properties is called distance-regular.","We deal first with distance-regular graphs with girth 3 and 4. We show any such graph which contains a cycle (v(,1),v(,2),v(,3),v(,4),v(,1)) where (PAR-DIFF)(v(,1),v(,3)) = (PAR-DIFF)(v(,2),v(,4)) = 2 is finite with diameter d bounded by its valency k. Next we obtain new "feasibility conditions" that the intersection numbers of arbitrary distance-regular graphs with girth 3 or 4 must satisfy. We use these conditions to generate the intersection array of certain distance-regular graphs from their valency and three other parameters.","We then study distance-regular graphs with arbitrary girth and extend the ideas used in the girth 3 or 4 case to obtain a bound on the diameter of a class of distance-regular graphs, including all those with even girth. We then introduce a generalization of a distance-regular graph called a (s,c,a,k)-graph, which possesses enough of the local structure of a distance-regular graph to enable us to find bounds on their diameters in some cases. Finally we obtain lower bounds on the eigenvalue multiplicities for distance-regular graphs in terms of their valence and girth, yielding additional feasibility conditions for their intersection arrays.","Made available in DSpace on 2014-12-16T06:18:03Z (GMT). No. of bitstreams: 1 8218574.pdf: 2612218 bytes, checksum: a2c414f514218467b97cd3b109618e49 (MD5) Previous issue date: 1982","Embargo set by: Seth Robbins for item 71370 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","115 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982."],"dc:identifier":["http://hdl.handle.net/2142/71204","(UMI)AAI8218574"],"dc:subject":["Mathematics"],"dc:title":["Distance-Regular Graphs and Generalizations"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}