{"id":"https://openalex.org/W7213952452","doi":"https://doi.org/10.48550/arxiv.2609.21529","title":"Spectral Extremal Graphs without a $K_k$-Factor","display_name":"Spectral Extremal Graphs without a $K_k$-Factor","publication_year":2026,"publication_date":"2026-09-18","ids":{"openalex":"https://openalex.org/W7213952452","doi":"https://doi.org/10.48550/arxiv.2609.21529"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.21529","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21529","pdf_url":"https://arxiv.org/pdf/2609.21529","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.21529","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005492982","display_name":"Cunxiang Duan","orcid":"https://orcid.org/0000-0003-2571-0212","observed_orcids":["https://orcid.org/0000-0003-2571-0212"]},"institutions":[{"id":"https://openalex.org/I143593769","display_name":"East China University of Science and Technology","ror":"https://ror.org/01vyrm377","country_code":"CN","type":"education","lineage":["https://openalex.org/I143593769"]},{"id":"https://openalex.org/I9086337","display_name":"Taiyuan University of Technology","ror":"https://ror.org/03kv08d37","country_code":"CN","type":"education","lineage":["https://openalex.org/I9086337"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Duan, Cunxiang","raw_affiliation_strings":["School of Mathematics , East China University of Science and Technology , Shanghai 200237 , P.R. China","School of Mathematics , Taiyuan University of Technology , Taiyuan , Shanxi 030024 , P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics , East China University of Science and Technology , Shanghai 200237 , P.R. China","institution_ids":["https://openalex.org/I143593769"]},{"raw_affiliation_string":"School of Mathematics , Taiyuan University of Technology , Taiyuan , Shanxi 030024 , P.R. China","institution_ids":["https://openalex.org/I9086337"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5152401460","display_name":"Tingting Han","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I94310126","display_name":"Shanxi Normal University","ror":"https://ror.org/03zd3ta61","country_code":"CN","type":"education","lineage":["https://openalex.org/I94310126"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Han, Tingting","raw_affiliation_strings":["School of Mathematical Sciences , Shanxi Normal University , Taiyuan , Shanxi 030031 , P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Shanxi Normal University , Taiyuan , Shanxi 030031 , P.R. China","institution_ids":["https://openalex.org/I94310126"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5152381782","display_name":"Lin-Peng Zhang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I154099455","display_name":"Shandong University","ror":"https://ror.org/0207yh398","country_code":"CN","type":"education","lineage":["https://openalex.org/I154099455"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhang, Lin-Peng","raw_affiliation_strings":["School of Mathematics , Shandong University , Jinan , Shandong 250100 , P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics , Shandong University , Jinan , Shandong 250100 , P.R. China","institution_ids":["https://openalex.org/I154099455"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9984999895095825,"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/T11476","display_name":"Graph theory and applications","score":0.9984999895095825,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.00039999998989515007,"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/T12292","display_name":"Graph Theory and Algorithms","score":9.999999747378752e-05,"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/extremal-graphs","display_name":"extremal graphs","score":0.8289999961853027},{"id":"https://openalex.org/keywords/forbidden-subgraphs","display_name":"forbidden subgraphs","score":0.5920000076293945},{"id":"https://openalex.org/keywords/rayleigh-quotient","display_name":"Rayleigh quotient","score":0.9340000152587891}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7993000149726868},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7601000070571899},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.5875999927520752},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.5442000031471252},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.5285000205039978},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5260999798774719},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4957999885082245},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4896000027656555},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.4390000104904175},{"id":"https://openalex.org/C74003402","wikidata":"https://www.wikidata.org/wiki/Q3180727","display_name":"Spectral graph theory","level":5,"score":0.39070001244544983},{"id":"https://openalex.org/C78913703","wikidata":"https://www.wikidata.org/wiki/Q5597087","display_name":"Graph energy","level":5,"score":0.34940001368522644},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.29589998722076416},{"id":"https://openalex.org/C2985906921","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Spectral properties","level":2,"score":0.27950000762939453},{"id":"https://openalex.org/C123482549","wikidata":"https://www.wikidata.org/wiki/Q826467","display_name":"Regular graph","level":5,"score":0.2727000117301941},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.26190000772476196}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.21529","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21529","pdf_url":"https://arxiv.org/pdf/2609.21529","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"doi:10.48550/arxiv.2609.21529","doi":"https://doi.org/10.48550/arxiv.2609.21529","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.21529","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.21529","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21529","pdf_url":"https://arxiv.org/pdf/2609.21529","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213952452.pdf","grobid_xml":"https://content.openalex.org/works/W7213952452.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4411689858","https://openalex.org/W4385890057","https://openalex.org/W3197431628","https://openalex.org/W4318313681","https://openalex.org/W4415191131","https://openalex.org/W4310425679","https://openalex.org/W4399664237","https://openalex.org/W2991359383","https://openalex.org/W3192159744","https://openalex.org/W4404045218"],"abstract_inverted_index":{"Let":[0],"$k\\ge":[1],"3$":[2],"and":[3,64,108,115],"let":[4],"$n=km$.":[5],"A":[6],"$K_k$-factor":[7,40],"in":[8],"an":[9,35],"$n$-vertex":[10,36],"graph":[11,37,51,73],"is":[12,74],"a":[13,93],"collection":[14],"of":[15,19,34,87],"$m$":[16],"vertex-disjoint":[17],"copies":[18],"$K_k$":[20],"that":[21,48],"covers":[22],"the":[23,29,70,88,101,105,116],"entire":[24],"vertex":[25,81],"set.":[26],"We":[27],"determine":[28],"maximum":[30],"adjacency":[31],"spectral":[32,109],"radius":[33],"containing":[38],"no":[39],"when":[41],"$m\\ge":[42],"2k-1$.":[43],"More":[44],"precisely,":[45],"we":[46],"prove":[47],"every":[49],"such":[50],"$G$":[52],"satisfies":[53],"\\[":[54],"\u03c1(G)\\le":[55],"\u03c1(H_{n,k}),":[56],"\\qquad":[57],"H_{n,k}=K_{k-2}\\vee\\bigl(K_{n-k+1}\\cup":[58],"K_1\\bigr),":[59],"\\]":[60],"with":[61,104],"equality":[62],"if":[63,66],"only":[65],"$G\\cong":[67],"H_{n,k}$.":[68],"Equivalently,":[69],"unique":[71],"extremal":[72],"obtained":[75],"from":[76,100],"$K_{n-1}$":[77],"by":[78],"adding":[79],"one":[80],"adjacent":[82],"to":[83],"exactly":[84],"$k-2$":[85],"vertices":[86],"clique.":[89],"Our":[90],"proof":[91],"combines":[92],"decomposition":[94],"lemma":[95],"for":[96],"sparse":[97],"complements,":[98],"derived":[99],"Hajnal--Szemer\u00e9di":[102],"theorem,":[103],"Motzkin--Straus":[106],"inequality":[107],"estimates":[110],"based":[111],"on":[112],"quotient":[113],"matrices":[114],"Rayleigh":[117],"quotient.":[118]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-22T00:00:00"}
