{"id":"https://openalex.org/W7212050206","doi":"https://doi.org/10.48550/arxiv.2609.06818","title":"The maximum relaxation time of a random walk on regular graphs","display_name":"The maximum relaxation time of a random walk on regular graphs","publication_year":2026,"publication_date":"2026-09-06","ids":{"openalex":"https://openalex.org/W7212050206","doi":"https://doi.org/10.48550/arxiv.2609.06818"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.06818","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.06818","pdf_url":"https://arxiv.org/pdf/2609.06818","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/2609.06818","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5136368988","display_name":"Haoran Zhu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zhu, Haoran","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5136368988"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":3,"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":0.8490999937057495,"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":0.8490999937057495,"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":0.042399998754262924,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.03280000016093254,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/random-walks-on-graphs","display_name":"random walks on graphs","score":0.7289999723434448},{"id":"https://openalex.org/keywords/relaxation-time","display_name":"relaxation time","score":0.9409999847412109},{"id":"https://openalex.org/keywords/regular-graphs","display_name":"regular graphs","score":0.9150000214576721},{"id":"https://openalex.org/keywords/algebraic-connectivity","display_name":"algebraic connectivity","score":0.7990000247955322},{"id":"https://openalex.org/keywords/graph-diameter","display_name":"graph diameter","score":0.5989999771118164},{"id":"https://openalex.org/keywords/edge-connectivity","display_name":"edge connectivity","score":0.8320000171661377},{"id":"https://openalex.org/keywords/first-passage-time","display_name":"first passage time","score":0.8059999942779541},{"id":"https://openalex.org/keywords/commuting-time","display_name":"commuting time","score":1.0}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8169999718666077},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5968999862670898},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5809999704360962},{"id":"https://openalex.org/C121194460","wikidata":"https://www.wikidata.org/wiki/Q856741","display_name":"Random walk","level":2,"score":0.5799999833106995},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.478300005197525},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.4562999904155731},{"id":"https://openalex.org/C181789720","wikidata":"https://www.wikidata.org/wiki/Q4812191","display_name":"Asymptotically optimal algorithm","level":2,"score":0.42719998955726624},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.42239999771118164},{"id":"https://openalex.org/C130432447","wikidata":"https://www.wikidata.org/wiki/Q11420049","display_name":"Quartic function","level":2,"score":0.41040000319480896},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4049000144004822},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3928999900817871},{"id":"https://openalex.org/C2778114796","wikidata":"https://www.wikidata.org/wiki/Q7575194","display_name":"Spectral gap","level":2,"score":0.3919000029563904},{"id":"https://openalex.org/C41949839","wikidata":"https://www.wikidata.org/wiki/Q1756677","display_name":"Stability theory","level":3,"score":0.38609999418258667},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.3837999999523163},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.3772999942302704},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.37310001254081726},{"id":"https://openalex.org/C4255713","wikidata":"https://www.wikidata.org/wiki/Q7291988","display_name":"Random regular graph","level":5,"score":0.3578999936580658},{"id":"https://openalex.org/C156387681","wikidata":"https://www.wikidata.org/wiki/Q6059484","display_name":"Invariance principle","level":2,"score":0.35499998927116394},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.33709999918937683},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.32670000195503235},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32249999046325684},{"id":"https://openalex.org/C2991992277","wikidata":"https://www.wikidata.org/wiki/Q473227","display_name":"Simple cubic lattice","level":3,"score":0.30250000953674316},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.2915000021457672},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.2881999909877777},{"id":"https://openalex.org/C20353970","wikidata":"https://www.wikidata.org/wiki/Q1056998","display_name":"Simple random sample","level":3,"score":0.2874999940395355},{"id":"https://openalex.org/C37727645","wikidata":"https://www.wikidata.org/wiki/Q6675770","display_name":"Loop-erased random walk","level":4,"score":0.2662000060081482},{"id":"https://openalex.org/C100500283","wikidata":"https://www.wikidata.org/wiki/Q1374495","display_name":"Cubic graph","level":5,"score":0.263700008392334},{"id":"https://openalex.org/C154547637","wikidata":"https://www.wikidata.org/wiki/Q776602","display_name":"Expander graph","level":3,"score":0.262800008058548},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.25369998812675476}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.06818","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.06818","pdf_url":"https://arxiv.org/pdf/2609.06818","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.2609.06818","doi":"https://doi.org/10.48550/arxiv.2609.06818","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.06818","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:2609.06818","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.06818","pdf_url":"https://arxiv.org/pdf/2609.06818","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/W7212050206.pdf","grobid_xml":"https://content.openalex.org/works/W7212050206.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7214115105","https://openalex.org/W7214121634","https://openalex.org/W3154694688","https://openalex.org/W3047823646","https://openalex.org/W2797354362","https://openalex.org/W4392908151","https://openalex.org/W7223015305","https://openalex.org/W4399973719","https://openalex.org/W2071531804","https://openalex.org/W4415779327"],"abstract_inverted_index":{"We":[0,41,74,136],"establish":[1,75,150],"sharp":[2,76,139,151,174],"quadratic":[3],"bounds":[4,77,141,152,175],"on":[5,13,78,176],"the":[6,49,57,88,93,154,173],"relaxation":[7],"time":[8],"of":[9,92,104,113,144,166],"simple":[10],"random":[11],"walk":[12],"connected":[14],"regular":[15,188],"graphs,":[16,148],"with":[17],"leading":[18],"constants":[19],"$3/(2\u03c0^2)$":[20],"and":[21,25,45,52,55,65,68,71,83,86,106,109,115,159,178,184],"$1/\u03c0^2$":[22],"for":[23,48,153,187],"even":[24],"odd":[26],"orders,":[27],"respectively.":[28],"This":[29,99],"resolves":[30,100],"a":[31,101,110,164],"longstanding":[32,102],"conjecture":[33,39,60,103,112,165],"known":[34],"as":[35],"Aldous--Fill":[36],"spectral":[37,161],"gap":[38,155],"(2002).":[40],"also":[42],"prove":[43,87,172],"uniqueness":[44,59],"stability":[46,120],"theorems":[47],"corresponding":[50],"cubic":[51],"quartic":[53,58],"chains,":[54],"settle":[56],"posed":[61],"by":[62,69,182],"Abdi,":[63],"Ghorbani,":[64],"Imrich":[66],"(2021)":[67],"Abdi":[70,114],"Ghorbani":[72,116],"(2023).":[73],"algebraic":[79,126],"connectivity":[80,127],"under":[81],"minimum-degree":[82],"regularity":[84],"constraints":[85],"conjectured":[89,181],"chain":[90],"structure":[91],"minimisers,":[94],"identifying":[95],"their":[96,133],"repeating":[97],"blocks.":[98],"Guiduli":[105],"Mohar":[107],"(1996)":[108],"structural":[111],"(2024).":[117,168],"Our":[118],"quantitative":[119],"theorem":[121],"shows":[122],"that":[123],"nearly":[124,129],"minimum":[125],"forces":[128],"maximum":[130,157],"diameter,":[131],"proving":[132],"diameter":[134],"conjecture.":[135],"further":[137],"obtain":[138],"relaxation-time":[140],"in":[142],"terms":[143],"edge-connectivity.":[145],"For":[146],"nonregular":[147],"we":[149,171],"between":[156],"degree":[158],"adjacency":[160],"radius,":[162],"confirming":[163],"Liu":[167],"As":[169],"applications,":[170],"hitting":[177],"commute":[179],"times":[180],"Aldous":[183],"Fill":[185],"(2002)":[186],"graphs.":[189]},"counts_by_year":[{"year":2026,"cited_by_count":3}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-10T00:00:00"}