{"id":"https://openalex.org/W2078321205","doi":"https://doi.org/10.1002/jgt.3190130411","title":"Lower bounds on the number of triangles in a graph","display_name":"Lower bounds on the number of triangles in a graph","publication_year":1989,"publication_date":"1989-09-01","ids":{"openalex":"https://openalex.org/W2078321205","doi":"https://doi.org/10.1002/jgt.3190130411","mag":"2078321205"},"language":"en","primary_location":{"is_oa":false,"landing_page_url":"https://doi.org/10.1002/jgt.3190130411","pdf_url":null,"source":{"id":"https://openalex.org/S193368155","display_name":"Journal of Graph Theory","issn_l":"0364-9024","issn":["0364-9024","1097-0118"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false},"type":"article","type_crossref":"journal-article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5077671733","display_name":"David Fisher","orcid":"https://orcid.org/0000-0003-2999-5518"},"institutions":[{"id":"https://openalex.org/I133543626","display_name":"Harvey Mudd College","ror":"https://ror.org/025ecfn45","country_code":"US","type":"education","lineage":["https://openalex.org/I133543626"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"David C. Fisher","raw_affiliation_strings":["Department of Mathematics, Harvey Mudd College, Claremont, California,#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Harvey Mudd College, Claremont, California,#TAB#","institution_ids":["https://openalex.org/I133543626"]}]}],"institution_assertions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5077671733"],"corresponding_institution_ids":["https://openalex.org/I133543626"],"apc_list":{"value":3760,"currency":"USD","value_usd":3760,"provenance":"doaj"},"apc_paid":null,"fwci":1.468,"has_fulltext":true,"fulltext_origin":"ngrams","cited_by_count":73,"citation_normalized_percentile":{"value":0.908108,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":96},"biblio":{"volume":"13","issue":"4","first_page":"505","last_page":"512"},"is_retracted":false,"is_paratext":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9996,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11476","display_name":"Graph theory and applications","score":0.9996,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9994,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.993,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/spectral-radius","display_name":"Spectral Radius","score":0.47990617},{"id":"https://openalex.org/keywords/complement","display_name":"Complement","score":0.4585102}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.89428663},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8254987},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5757506},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.57464445},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.47990617},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.4585102},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.11457163},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.082464725},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.055050433},{"id":"https://openalex.org/C127716648","wikidata":"https://www.wikidata.org/wiki/Q104053","display_name":"Phenotype","level":3,"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/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"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/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"is_oa":false,"landing_page_url":"https://doi.org/10.1002/jgt.3190130411","pdf_url":null,"source":{"id":"https://openalex.org/S193368155","display_name":"Journal of Graph Theory","issn_l":"0364-9024","issn":["0364-9024","1097-0118"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false}],"best_oa_location":null,"sustainable_development_goals":[],"grants":[],"datasets":[],"versions":[],"referenced_works_count":7,"referenced_works":["https://openalex.org/W1510730180","https://openalex.org/W2049005199","https://openalex.org/W2079412785","https://openalex.org/W2115164934","https://openalex.org/W2312510043","https://openalex.org/W2914085271","https://openalex.org/W4210766026"],"related_works":["https://openalex.org/W4293788411","https://openalex.org/W3148147474","https://openalex.org/W2963001570","https://openalex.org/W2962954592","https://openalex.org/W2907203903","https://openalex.org/W2382566571","https://openalex.org/W2369247774","https://openalex.org/W2318654273","https://openalex.org/W2274753335","https://openalex.org/W1602350308"],"abstract_inverted_index":{"Abstract":[0],"For":[1],"a":[2,109,147,165,178],"simple":[3],"graph":[4,179],"G":[5,43,48,75,81,118],",":[6],"let":[7],"f":[8,59],"(":[9,47,60,80,190],"z":[10,17,21,26,61],")":[11,49,66,82,86,142,193],"\u2261":[12],"1":[13,16,98,106,127,132,137,182,210,214],"\u2212":[14,23,128,133,138,199,211,215],"c":[15,19,24,31,96,101,104,120,124,126,130,135,140,181,185,191,201,205,208,213],"+":[18,28,87,203],"3":[20,25,27,121,139,192,200,209],"2":[22,97,102,105,125,129,131,136,141,186,197,198,202,206],"\u2026":[29],"where":[30],"k":[32,40],"is":[33,90,119,152,188],"the":[34,51,54,68,72,113,172,222],"number":[35,114,174],"of":[36,53,58,71,74,115,149,164,175],"complete":[37],"subgraphs":[38],"on":[39,112],"nodes":[41,183],"in":[42,117,177],".":[44,76],"Let":[45,63],"r":[46,79],"be":[50,67],"reciprocal":[52],"smallest":[55],"real":[56],"root":[57],").":[62],"\u03bb(":[64,84],"\u1e20":[65,85],"spectral":[69],"radius":[70],"complement":[73],"We":[77],"show":[78,93],"\u2a7e":[83,122],"1.":[88],"This":[89,145],"used":[91],"to":[92,221],"that":[94,159],"if":[95],"/4":[99],"\u2a7d":[100,103],"/3,":[107],"then":[108],"lower":[110,225],"bound":[111,148],"triangles":[116,176],"[9":[123],"2(":[134],"3/2":[143],"]/27.":[144],"improves":[146],"Bollob\u00e1s":[150],"and":[151,169,171,184],"asymptotically":[153],"sharp.":[154],"Also,":[155],"this":[156],"paper":[157],"shows":[158],"magnified":[160],"image":[161],"(a":[162],"corollary":[163],"result":[166],"from":[167],"Erd\u00f6s":[168],"Hanini)":[170],"average":[173],"with":[180],"edges":[187],"E":[189],"=":[194],"3/4(":[195],"C":[196],"2)":[204],"/(":[207],"5":[212],"4).":[216],"These":[217],"are":[218],"graphically":[219],"compared":[220],"best":[223],"known":[224],"bounds.":[226]},"cited_by_api_url":"https://api.openalex.org/works?filter=cites:W2078321205","counts_by_year":[{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":8},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":7},{"year":2012,"cited_by_count":5}],"updated_date":"2025-01-06T14:41:01.049952","created_date":"2016-06-24"}