{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71261"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71261","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Automorphism Groups of the Augmented Distance Graphs of Trees","abstract":"In algebraic graph theory one studies algebraic variants of graphs by forming matrices and groups relating to the graph. One example of this is the distance matrices, $\\Gamma\\sb{\\rm i}$, and their associated groups.","abstract_html":"In algebraic graph theory one studies algebraic variants of graphs by forming matrices and groups relating to the graph. One example of this is the distance matrices, $\\Gamma\\sb{\\rm i}$, and their associated groups.","abstract_has_math":true,"creators":["Sportsman, Joseph Scott"],"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":"2014-12-16T06:18:21Z","date_published":"2014-12-16T06:18:21Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8803208"],"render_values":[{"text":"(UMI)AAI8803208","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71261","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Sportsman, Joseph Scott"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:21Z","10000-01-01","1987"]},{"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/2142/71261","(UMI)AAI8803208"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In algebraic graph theory one studies algebraic variants of graphs by forming matrices and groups relating to the graph. One example of this is the distance matrices, $\\Gamma\\sb{\\rm i}$, and their associated groups.","In this thesis we introduce the graphs, $\\Gamma\\sp{\\rm (r)}$ defined by $\\Gamma\\sp{\\rm (r)}$ = $\\Gamma\\sb1$ + $\\Gamma\\sb2$ + $\\cdots$ + $\\Gamma\\sb{\\rm r}$ and their automorphism groups G$\\sp{\\rm (r)}$. We show that for a tree $\\Gamma$, the groups G$\\sp{\\rm (r)}$ form a tower which is not the case for arbitrary graphs. From this, we give a description of the structure of G$\\sp{\\rm (r)}$ for trees and completely characterize the trees of a fixed diameter which have minimal group tower length. Also we introduce a new parameter, $\\chi$ for trees defined as follows: Let x and y be vertices of $\\Gamma$. Partition the remaining vertices into three sets; W(x) = $\\{$w$\\epsilon$V($\\Gamma$): $\\partial$(w,x)$$ 0$\\}$. It turns out that $\\chi$ has nice properties. One theorem we prove is the following: If $\\Gamma$ is a tree of diameter greater than 3, and m = min$\\{\\chi$ + 1, (d/2) $\\}$, then G$\\sp{\\rm (m+1)}$ $\\not=$ G, but G$\\sp{\\rm (r)}$ = G for all r $\\leq$ m.","Made available in DSpace on 2014-12-16T06:18:21Z (GMT). No. of bitstreams: 1 8803208.pdf: 2627812 bytes, checksum: 59b3c0be0e38c308a2158e2ab4b8ad13 (MD5) Previous issue date: 1987","Embargo set by: Seth Robbins for item 71427 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","92 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987."]},{"key":"dc:title","label":"Title","values":["Automorphism Groups of the Augmented Distance Graphs of Trees"]}]}],"canonical_facts":{"dc:creator":["Sportsman, Joseph Scott"],"dc:date":["2014-12-16T06:18:21Z","10000-01-01","1987"],"dc:description":["In algebraic graph theory one studies algebraic variants of graphs by forming matrices and groups relating to the graph. One example of this is the distance matrices, $\\Gamma\\sb{\\rm i}$, and their associated groups.","In this thesis we introduce the graphs, $\\Gamma\\sp{\\rm (r)}$ defined by $\\Gamma\\sp{\\rm (r)}$ = $\\Gamma\\sb1$ + $\\Gamma\\sb2$ + $\\cdots$ + $\\Gamma\\sb{\\rm r}$ and their automorphism groups G$\\sp{\\rm (r)}$. We show that for a tree $\\Gamma$, the groups G$\\sp{\\rm (r)}$ form a tower which is not the case for arbitrary graphs. From this, we give a description of the structure of G$\\sp{\\rm (r)}$ for trees and completely characterize the trees of a fixed diameter which have minimal group tower length. Also we introduce a new parameter, $\\chi$ for trees defined as follows: Let x and y be vertices of $\\Gamma$. Partition the remaining vertices into three sets; W(x) = $\\{$w$\\epsilon$V($\\Gamma$): $\\partial$(w,x)$$ 0$\\}$. It turns out that $\\chi$ has nice properties. One theorem we prove is the following: If $\\Gamma$ is a tree of diameter greater than 3, and m = min$\\{\\chi$ + 1, (d/2) $\\}$, then G$\\sp{\\rm (m+1)}$ $\\not=$ G, but G$\\sp{\\rm (r)}$ = G for all r $\\leq$ m.","Made available in DSpace on 2014-12-16T06:18:21Z (GMT). No. of bitstreams: 1 8803208.pdf: 2627812 bytes, checksum: 59b3c0be0e38c308a2158e2ab4b8ad13 (MD5) Previous issue date: 1987","Embargo set by: Seth Robbins for item 71427 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","92 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987."],"dc:identifier":["http://hdl.handle.net/2142/71261","(UMI)AAI8803208"],"dc:subject":["Mathematics"],"dc:title":["Automorphism Groups of the Augmented Distance Graphs of Trees"],"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"}