{"id":"https://openalex.org/W3081486720","doi":"https://doi.org/10.37236/9802","title":"The Non-Commuting, Non-Generating Graph of a Nilpotent Group","display_name":"The Non-Commuting, Non-Generating Graph of a Nilpotent Group","publication_year":2021,"publication_date":"2021-01-29","ids":{"openalex":"https://openalex.org/W3081486720","doi":"https://doi.org/10.37236/9802","mag":"3081486720"},"language":"en","primary_location":{"id":"doi:10.37236/9802","is_oa":true,"landing_page_url":"https://doi.org/10.37236/9802","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v28i1p16/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v28i1p16/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048639170","display_name":"Peter J\u200e. Cameron","orcid":"https://orcid.org/0000-0003-3130-9505"},"institutions":[{"id":"https://openalex.org/I16835326","display_name":"University of St Andrews","ror":"https://ror.org/02wn5qz54","country_code":"GB","type":"education","lineage":["https://openalex.org/I16835326"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Peter Cameron","raw_affiliation_strings":["University of St Andrews"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of St Andrews","institution_ids":["https://openalex.org/I16835326"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5051756142","display_name":"Saul D. Freedman","orcid":"https://orcid.org/0000-0001-8062-0549"},"institutions":[{"id":"https://openalex.org/I16835326","display_name":"University of St Andrews","ror":"https://ror.org/02wn5qz54","country_code":"GB","type":"education","lineage":["https://openalex.org/I16835326"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Saul Freedman","raw_affiliation_strings":["University of St Andrews"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of St Andrews","institution_ids":["https://openalex.org/I16835326"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5030261802","display_name":"Colva M. Roney\u2010Dougal","orcid":"https://orcid.org/0000-0002-0532-3349"},"institutions":[{"id":"https://openalex.org/I16835326","display_name":"University of St Andrews","ror":"https://ror.org/02wn5qz54","country_code":"GB","type":"education","lineage":["https://openalex.org/I16835326"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Colva Roney-Dougal","raw_affiliation_strings":["University of St Andrews"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of St Andrews","institution_ids":["https://openalex.org/I16835326"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I16835326"],"apc_list":null,"apc_paid":null,"fwci":4.4104,"has_fulltext":true,"cited_by_count":9,"citation_normalized_percentile":{"value":0.94479102,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":93,"max":97},"biblio":{"volume":"28","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8610947132110596},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6833534240722656},{"id":"https://openalex.org/keywords/nilpotent","display_name":"Nilpotent","score":0.6815295219421387},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6613032817840576},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.6116751432418823},{"id":"https://openalex.org/keywords/complement","display_name":"Complement (music)","score":0.49993228912353516},{"id":"https://openalex.org/keywords/bound-graph","display_name":"Bound graph","score":0.414654403924942},{"id":"https://openalex.org/keywords/graph-power","display_name":"Graph power","score":0.2647975981235504},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.06218269467353821},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.055799245834350586}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8610947132110596},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6833534240722656},{"id":"https://openalex.org/C50555996","wikidata":"https://www.wikidata.org/wiki/Q840023","display_name":"Nilpotent","level":2,"score":0.6815295219421387},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6613032817840576},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.6116751432418823},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.49993228912353516},{"id":"https://openalex.org/C121134508","wikidata":"https://www.wikidata.org/wiki/Q4949882","display_name":"Bound graph","level":5,"score":0.414654403924942},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.2647975981235504},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.06218269467353821},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.055799245834350586},{"id":"https://openalex.org/C127716648","wikidata":"https://www.wikidata.org/wiki/Q104053","display_name":"Phenotype","level":3,"score":0.0},{"id":"https://openalex.org/C188082640","wikidata":"https://www.wikidata.org/wiki/Q1780899","display_name":"Complementation","level":4,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/9802","is_oa":true,"landing_page_url":"https://doi.org/10.37236/9802","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v28i1p16/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2008.09291","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2008.09291","pdf_url":"https://arxiv.org/pdf/2008.09291","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:research-repository.st-andrews.ac.uk:10023/21413","is_oa":true,"landing_page_url":"https://hdl.handle.net/10023/21413","pdf_url":null,"source":{"id":"https://openalex.org/S4306400230","display_name":"St Andrews Research Repository (St Andrews Research Repository)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I16835326","host_organization_name":"University of St Andrews","host_organization_lineage":["https://openalex.org/I16835326"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nd","license_id":"https://openalex.org/licenses/cc-by-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Journal article"}],"best_oa_location":{"id":"doi:10.37236/9802","is_oa":true,"landing_page_url":"https://doi.org/10.37236/9802","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v28i1p16/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2189360572","display_name":null,"funder_award_id":"EP/R014604/1","funder_id":"https://openalex.org/F4320313284","funder_display_name":"Isaac Newton Institute for Mathematical Sciences"},{"id":"https://openalex.org/G3691759950","display_name":null,"funder_award_id":"EP/R014604/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320306164","display_name":"Simons Foundation","ror":"https://ror.org/01cmst727"},{"id":"https://openalex.org/F4320313284","display_name":"Isaac Newton Institute for Mathematical Sciences","ror":null},{"id":"https://openalex.org/F4320320278","display_name":"University of St Andrews","ror":"https://ror.org/02wn5qz54"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3081486720.pdf","grobid_xml":"https://content.openalex.org/works/W3081486720.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W1649168892","https://openalex.org/W1964770012","https://openalex.org/W1966831799","https://openalex.org/W1976677460","https://openalex.org/W2003004532","https://openalex.org/W2017573367","https://openalex.org/W2019945630","https://openalex.org/W2043225560","https://openalex.org/W2048857224","https://openalex.org/W2055566058","https://openalex.org/W2081545579","https://openalex.org/W2091688497","https://openalex.org/W2116802568","https://openalex.org/W2225205306","https://openalex.org/W2302395355","https://openalex.org/W2319687064","https://openalex.org/W2589708980","https://openalex.org/W2594692962","https://openalex.org/W2951493460","https://openalex.org/W2963083310","https://openalex.org/W2963403635","https://openalex.org/W3033383352","https://openalex.org/W4213060864","https://openalex.org/W4239653429","https://openalex.org/W4394642779","https://openalex.org/W6654573219"],"related_works":["https://openalex.org/W3033929256","https://openalex.org/W2226543729","https://openalex.org/W2937406279","https://openalex.org/W4381094520","https://openalex.org/W1527576459","https://openalex.org/W2809930434","https://openalex.org/W2793144387","https://openalex.org/W3159927204","https://openalex.org/W1990117560","https://openalex.org/W2157163513"],"abstract_inverted_index":{"For":[0],"a":[1],"nilpotent":[2],"group":[3,110],"$G$,":[4,24],"let":[5,61],"$\\Xi(G)$":[6,36,67,77,130],"be":[7,63],"the":[8,11,14,20,64,95,100,122,125],"difference":[9],"between":[10,124],"complement":[12],"of":[13,17,23,31,66,127],"generating":[15],"graph":[16,22],"$G$":[18,32,117,128],"and":[19,48,55,129],"commuting":[21],"with":[25,43,85,90,111],"vertices":[26,45],"corresponding":[27],"to":[28,108],"central":[29],"elements":[30],"removed.":[33],"That":[34],"is,":[35],"has":[37,78],"vertex":[38],"set":[39],"$G":[40],"\\setminus":[41],"Z(G)$,":[42],"two":[44],"adjacent":[46],"if":[47,50,76],"only":[49],"they":[51],"do":[52,56],"not":[53,57],"commute":[54],"generate":[58],"$G$.":[59],"Additionally,":[60],"$\\Xi^+(G)$":[62,82],"subgraph":[65],"induced":[68],"by":[69],"its":[70],"non-isolated":[71],"vertices.":[72],"We":[73],"show":[74],"that":[75],"an":[79],"edge,":[80],"then":[81],"is":[83,118],"connected":[84],"diameter":[86,96],"$2$":[87],"or":[88],"$3$,":[89],"$\\Xi(G)":[91],"=":[92],"\\Xi^+(G)$":[93],"in":[94,131],"$3$":[97],"case.":[98],"In":[99],"infinite":[101],"case,":[102],"our":[103],"results":[104],"apply":[105],"more":[106,132],"generally,":[107],"any":[109],"every":[112],"maximal":[113],"subgroup":[114],"normal.":[115],"When":[116],"finite,":[119],"we":[120],"explore":[121],"relationship":[123],"structures":[126],"detail.":[133]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
