{"id":"https://openalex.org/W7213929356","doi":"https://doi.org/10.48550/arxiv.2609.21367","title":"A solution to a conjecture on the signless Laplacian spectral radius for $t$-color-critical graphs","display_name":"A solution to a conjecture on the signless Laplacian spectral radius for $t$-color-critical graphs","publication_year":2026,"publication_date":"2026-09-18","ids":{"openalex":"https://openalex.org/W7213929356","doi":"https://doi.org/10.48550/arxiv.2609.21367"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.21367","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21367","pdf_url":"https://arxiv.org/pdf/2609.21367","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/2609.21367","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5125961467","display_name":"Ming-Zhu Chen","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I20942203","display_name":"Hainan University","ror":"https://ror.org/03q648j11","country_code":"CN","type":"education","lineage":["https://openalex.org/I20942203"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Chen, Ming-Zhu","raw_affiliation_strings":["School of Mathematics and Statistics , Hainan University , Haikou 570228 , P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics , Hainan University , Haikou 570228 , P.R. China","institution_ids":["https://openalex.org/I20942203"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065842793","display_name":"Ya-Lei Jin","orcid":"https://orcid.org/0000-0001-9415-2678","observed_orcids":["https://orcid.org/0000-0001-9415-2678"]},"institutions":[{"id":"https://openalex.org/I21945476","display_name":"Shanghai Normal University","ror":"https://ror.org/01cxqmw89","country_code":"CN","type":"education","lineage":["https://openalex.org/I21945476"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jin, Ya-Lei","raw_affiliation_strings":["Department of Mathematics , Shanghai Normal University , Shanghai 200234 , P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Shanghai Normal University , Shanghai 200234 , P.R. China","institution_ids":["https://openalex.org/I21945476"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5125964264","display_name":"Peng-Li Zhang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I905225518","display_name":"Shanghai University of International Business and Economics","ror":"https://ror.org/031t68441","country_code":"CN","type":"education","lineage":["https://openalex.org/I905225518"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhang, Peng-Li","raw_affiliation_strings":["School of Statistics and Data Science , Shanghai University of International Business and Economics , Shanghai 201620 , P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Statistics and Data Science , Shanghai University of International Business and Economics , Shanghai 201620 , P.R. China","institution_ids":["https://openalex.org/I905225518"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5152357512","display_name":"Jian Zheng","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I177739611","display_name":"Yangtze University","ror":"https://ror.org/05bhmhz54","country_code":"CN","type":"education","lineage":["https://openalex.org/I177739611"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zheng, Jian","raw_affiliation_strings":["School of Information and Mathematics , Yangtze University , Jingzhou 434023 , P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Information and Mathematics , Yangtze University , Jingzhou 434023 , P.R. China","institution_ids":["https://openalex.org/I177739611"]}]}],"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":1,"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":0.9976000189781189,"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":0.9976000189781189,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.0007999999797903001,"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"}},{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.0003000000142492354,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/signless-laplacian-spectral-radius","display_name":"signless Laplacian spectral radius","score":0.984000027179718},{"id":"https://openalex.org/keywords/induced-matchings","display_name":"induced matchings","score":0.6759999990463257},{"id":"https://openalex.org/keywords/f-free-graphs","display_name":"F-free graphs","score":0.7350000143051147},{"id":"https://openalex.org/keywords/extremal-graph-theory","display_name":"extremal graph theory","score":0.7900000214576721},{"id":"https://openalex.org/keywords/perron-vector","display_name":"Perron vector","score":0.9300000071525574},{"id":"https://openalex.org/keywords/chromatic-number","display_name":"chromatic number","score":0.7229999899864197},{"id":"https://openalex.org/keywords/join-of-graphs","display_name":"join of graphs","score":0.4880000054836273}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8513000011444092},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7872999906539917},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.7603999972343445},{"id":"https://openalex.org/C64999384","wikidata":"https://www.wikidata.org/wiki/Q3507967","display_name":"Symmetrization","level":2,"score":0.6705999970436096},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6395999789237976},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.541100025177002},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.46959999203681946},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4458000063896179},{"id":"https://openalex.org/C115178988","wikidata":"https://www.wikidata.org/wiki/Q772067","display_name":"Laplacian matrix","level":3,"score":0.43939998745918274},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.43540000915527344},{"id":"https://openalex.org/C7948225","wikidata":"https://www.wikidata.org/wiki/Q4723998","display_name":"Algebraic connectivity","level":4,"score":0.43389999866485596},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.41370001435279846},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.37290000915527344},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.31459999084472656},{"id":"https://openalex.org/C2993105083","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Simple graph","level":3,"score":0.2700999975204468},{"id":"https://openalex.org/C74003402","wikidata":"https://www.wikidata.org/wiki/Q3180727","display_name":"Spectral graph theory","level":5,"score":0.25920000672340393}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.21367","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21367","pdf_url":"https://arxiv.org/pdf/2609.21367","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.2609.21367","doi":"https://doi.org/10.48550/arxiv.2609.21367","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.21367","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:2609.21367","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.21367","pdf_url":"https://arxiv.org/pdf/2609.21367","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":[{"id":"https://openalex.org/G1413495475","display_name":null,"funder_award_id":"12371354","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2966349816","display_name":null,"funder_award_id":"12471331","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4511811078","display_name":null,"funder_award_id":"12461063","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5123942828","display_name":null,"funder_award_id":"12501493","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G561753478","display_name":null,"funder_award_id":"12371349","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213929356.pdf","grobid_xml":"https://content.openalex.org/works/W7213929356.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W7076454294","https://openalex.org/W7128790344","https://openalex.org/W7166662109"],"related_works":["https://openalex.org/W6891884712","https://openalex.org/W7166662109","https://openalex.org/W7128790344","https://openalex.org/W2969118961","https://openalex.org/W2415989501","https://openalex.org/W3033694756","https://openalex.org/W7213473557","https://openalex.org/W7138942476","https://openalex.org/W2093376397","https://openalex.org/W4391262717"],"abstract_inverted_index":{"An":[0],"induced":[1,9,18],"matching":[2,5,19],"is":[3,13,93,127,141],"a":[4,37,120,162,170],"that":[6],"forms":[7],"an":[8,128,142],"subgraph.":[10],"A":[11],"graph":[12,39,52,167],"$t$-color-critical":[14,38],"if":[15],"removing":[16,28],"some":[17],"of":[20,112,119,130,157,161],"size":[21],"$t$":[22],"lowers":[23],"its":[24],"chromatic":[25],"number,":[26],"but":[27],"any":[29],"$t-1$":[30],"vertices":[31],"does":[32],"not.":[33],"Let":[34],"$F$":[35],"be":[36],"with":[40],"$\u03c7(F)=r+1$.":[41],"For":[42],"sufficiently":[43,94],"large":[44],"$n$,":[45,145],"Simonovits":[46],"determined":[47],"the":[48,75,81,106,134,150,154,158],"unique":[49],"edge-extremal":[50],"$F$-free":[51,89],"on":[53,116,144,153],"$n$":[54,92],"vertices.":[55],"Recently,":[56],"Zheng,":[57],"Li":[58,60],"and":[59,72],"[Linear":[61],"Algebra":[62],"Appl.\\":[63],"730":[64],"(2026)":[65],"546--565]":[66],"conjectured":[67],"that,":[68],"for":[69,133],"$t\\ge":[70],"2$":[71],"$r\\ge":[73],"3$,":[74],"join":[76],"$K_{t-1}\\vee":[77],"T_{n-t+1,r}$":[78],"uniquely":[79],"maximizes":[80],"signless":[82,135,163],"Laplacian":[83,136,164],"spectral":[84,108,165],"radius":[85],"among":[86],"all":[87],"$n$-vertex":[88],"graphs":[90],"when":[91],"large.":[95],"In":[96,103],"this":[97,101,113],"paper,":[98],"we":[99,148],"prove":[100],"conjecture.":[102],"contrast":[104],"to":[105],"usual":[107],"arguments,":[109],"our":[110],"proof":[111],"conjecture":[114],"relies":[115],"two":[117],"techniques":[118],"rather":[121,168],"different":[122],"flavour.":[123],"Our":[124,138],"first":[125],"technique":[126,140],"analogue":[129],"Zykov":[131],"symmetrization":[132],"matrix.":[137],"second":[139],"induction":[143],"from":[146],"which":[147],"obtain":[149],"lower":[151],"bound":[152],"smallest":[155],"entry":[156],"Perron":[159],"vector":[160],"extremal":[166],"than":[169],"structural":[171],"statement.":[172]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-22T00:00:00"}
