{"id":"https://openalex.org/W7220413683","doi":"https://doi.org/10.48550/arxiv.2610.02913","title":"A finite threshold for the double-comet conjecture","display_name":"A finite threshold for the double-comet conjecture","publication_year":2026,"publication_date":"2026-10-02","ids":{"openalex":"https://openalex.org/W7220413683","doi":"https://doi.org/10.48550/arxiv.2610.02913"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.02913","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02913","pdf_url":"https://arxiv.org/pdf/2610.02913","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/2610.02913","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5140247196","display_name":"Dongxiu Cai","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Cai, Dongxiu","raw_affiliation_strings":["School of Mathematical Sciences , MOE-LSC , SHL-MAC , Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240 , P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , MOE-LSC , SHL-MAC , Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240 , P.R. China","institution_ids":["https://openalex.org/I183067930"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5151366926","display_name":"Zhenbo Chen","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Chen, Zhenbo","raw_affiliation_strings":["School of Mathematical Sciences , MOE-LSC , SHL-MAC , Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240 , P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , MOE-LSC , SHL-MAC , Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240 , P.R. China","institution_ids":["https://openalex.org/I183067930"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5157616849","display_name":"Xiao-Dong Zhang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhang, Xiao-Dong","raw_affiliation_strings":["School of Mathematical Sciences , MOE-LSC , SHL-MAC , Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240 , P.R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , MOE-LSC , SHL-MAC , Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240 , P.R. China","institution_ids":["https://openalex.org/I183067930"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I183067930"],"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.0007999999797903001,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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":0.00039999998989515007,"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/trees","display_name":"trees","score":0.45100000500679016},{"id":"https://openalex.org/keywords/effective-resistance","display_name":"effective resistance","score":0.8309999704360962},{"id":"https://openalex.org/keywords/dirichlet-forms","display_name":"Dirichlet forms","score":0.6610000133514404},{"id":"https://openalex.org/keywords/schur-complement","display_name":"Schur complement","score":0.9259999990463257},{"id":"https://openalex.org/keywords/adjacency-eigenvalues","display_name":"adjacency matrix eigenvalues","score":0.5809999704360962}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8737999796867371},{"id":"https://openalex.org/C2778114796","wikidata":"https://www.wikidata.org/wiki/Q7575194","display_name":"Spectral gap","level":2,"score":0.76910001039505},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6976000070571899},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6003999710083008},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.534500002861023},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.5177000164985657},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.5139999985694885},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.5080000162124634},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.423799991607666},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.4221000075340271},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4154999852180481},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.3732999861240387},{"id":"https://openalex.org/C199277547","wikidata":"https://www.wikidata.org/wiki/Q5872687","display_name":"Hitting time","level":2,"score":0.36469998955726624},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.35190001130104065},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.3465000092983246},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.34209999442100525},{"id":"https://openalex.org/C30014739","wikidata":"https://www.wikidata.org/wiki/Q5159445","display_name":"Cone (formal languages)","level":2,"score":0.33709999918937683},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.25130000710487366}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.02913","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02913","pdf_url":"https://arxiv.org/pdf/2610.02913","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.2610.02913","doi":"https://doi.org/10.48550/arxiv.2610.02913","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.02913","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.02913","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02913","pdf_url":"https://arxiv.org/pdf/2610.02913","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220413683.pdf","grobid_xml":"https://content.openalex.org/works/W7220413683.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2760156479","https://openalex.org/W7214944592","https://openalex.org/W2967948787","https://openalex.org/W4410070236","https://openalex.org/W3123327511","https://openalex.org/W3161370521","https://openalex.org/W2904968428","https://openalex.org/W2043694962","https://openalex.org/W2088296537","https://openalex.org/W7206186767"],"abstract_inverted_index":{"For":[0,112],"a":[1,31,40,64],"tree":[2,41,59],"$T$,":[3],"let":[4],"$g(T)=\u03bb_1(T)-\u03bb_2(T)$":[5],"be":[6],"the":[7,27,43,80,86,93,103],"difference":[8],"between":[9,89],"its":[10],"two":[11,28,90],"largest":[12],"adjacency":[13,45],"eigenvalues.":[14],"A":[15],"balanced":[16,65],"double":[17,66],"comet":[18],"is":[19,63],"obtained":[20],"by":[21,100],"attaching":[22],"equally":[23],"many":[24],"leaves":[25],"to":[26],"endpoints":[29],"of":[30,50,60,96],"path.":[32],"Jovovi\u0107,":[33],"Koledin":[34],"and":[35],"Stani\u0107":[36],"conjectured":[37],"that":[38,56],"such":[39],"attains":[42],"minimum":[44],"spectral":[46,82],"gap":[47,83,106],"among":[48],"trees":[49],"any":[51,73],"fixed":[52],"order.":[53],"We":[54,68],"prove":[55],"every":[57],"minimizing":[58],"order":[61],"$n\\ge200$":[62],"comet.":[67],"also":[69],"show":[70],"that,":[71],"for":[72,107,122],"finite":[74],"irreducible":[75],"reversible":[76],"continuous-time":[77],"Markov":[78],"chain,":[79],"inverse":[81,104],"differs":[84],"from":[85],"effective":[87],"resistance":[88],"states":[91],"times":[92],"stationary":[94],"variance":[95],"their":[97],"hitting":[98],"probability":[99],"at":[101,109],"most":[102],"Dirichlet":[105],"killing":[108],"those":[110],"states.":[111],"Perron":[113],"chains,":[114],"we":[115],"give":[116],"an":[117],"exact":[118],"two-vertex":[119],"Schur-complement":[120],"formula":[121],"this":[123],"resistance--variance":[124],"quantity.":[125]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-10-06T00:00:00"}