{"id":"https://openalex.org/W7204226322","doi":"https://doi.org/10.48550/arxiv.2608.24754","title":"Structure theorems for Lichnerowicz-sharp graphs","display_name":"Structure theorems for Lichnerowicz-sharp graphs","publication_year":2026,"publication_date":"2026-08-25","ids":{"openalex":"https://openalex.org/W7204226322","doi":"https://doi.org/10.48550/arxiv.2608.24754"},"language":null,"primary_location":{"id":"pmh:oai:arXiv.org:2608.24754","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24754","pdf_url":"https://arxiv.org/pdf/2608.24754","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.24754","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5138211394","display_name":"Yanlong Ding","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I126520041","display_name":"University of Science and Technology of China","ror":"https://ror.org/04c4dkn09","country_code":"CN","type":"education","lineage":["https://openalex.org/I126520041","https://openalex.org/I19820366"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ding, Yanlong","raw_affiliation_strings":["School of Mathematical Sciences , University of Science and Technology of China , Hefei 230026 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , University of Science and Technology of China , Hefei 230026 , China","institution_ids":["https://openalex.org/I126520041"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101517415","display_name":"Shiping Liu","orcid":"https://orcid.org/0000-0002-9987-1935","observed_orcids":["https://orcid.org/0000-0002-9987-1935"]},"institutions":[{"id":"https://openalex.org/I126520041","display_name":"University of Science and Technology of China","ror":"https://ror.org/04c4dkn09","country_code":"CN","type":"education","lineage":["https://openalex.org/I126520041","https://openalex.org/I19820366"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Liu, Shiping","raw_affiliation_strings":["School of Mathematical Sciences , University of Science and Technology of China , Hefei 230026 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , University of Science and Technology of China , Hefei 230026 , China","institution_ids":["https://openalex.org/I126520041"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5138217538","display_name":"Chiyu Zhou","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I126520041","display_name":"University of Science and Technology of China","ror":"https://ror.org/04c4dkn09","country_code":"CN","type":"education","lineage":["https://openalex.org/I126520041","https://openalex.org/I19820366"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhou, Chiyu","raw_affiliation_strings":["School of Mathematical Sciences , University of Science and Technology of China , Hefei 230026 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , University of Science and Technology of China , Hefei 230026 , China","institution_ids":["https://openalex.org/I126520041"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I126520041"],"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9810000061988831,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9810000061988831,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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":0.014100000262260437,"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/T11476","display_name":"Graph theory and applications","score":0.000699999975040555,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/hypercube-graphs","display_name":"hypercube graphs","score":0.7059999704360962},{"id":"https://openalex.org/keywords/bakry-emery-curvature","display_name":"Bakry-\u00c9mery curvature","score":0.9129999876022339},{"id":"https://openalex.org/keywords/first-eigenvalue","display_name":"first eigenvalue","score":0.4490000009536743}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8208000063896179},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6274999976158142},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5209000110626221},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5116999745368958},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5029000043869019},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.4625000059604645},{"id":"https://openalex.org/C50820777","wikidata":"https://www.wikidata.org/wiki/Q213723","display_name":"Hypercube","level":2,"score":0.445499986410141},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4300999939441681},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.4163999855518341},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.3813999891281128},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.35499998927116394},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.3513000011444092},{"id":"https://openalex.org/C74133993","wikidata":"https://www.wikidata.org/wiki/Q3115472","display_name":"Indifference graph","level":3,"score":0.3472999930381775},{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.3379000127315521},{"id":"https://openalex.org/C14201764","wikidata":"https://www.wikidata.org/wiki/Q1205074","display_name":"Symmetric graph","level":5,"score":0.3255999982357025},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.31769999861717224},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.3098999857902527},{"id":"https://openalex.org/C123482549","wikidata":"https://www.wikidata.org/wiki/Q826467","display_name":"Regular graph","level":5,"score":0.3068000078201294},{"id":"https://openalex.org/C2778134712","wikidata":"https://www.wikidata.org/wiki/Q1047307","display_name":"Bundle","level":2,"score":0.30059999227523804},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.29980000853538513},{"id":"https://openalex.org/C68103157","wikidata":"https://www.wikidata.org/wiki/Q569347","display_name":"Graph product","level":5,"score":0.29429998993873596},{"id":"https://openalex.org/C64339825","wikidata":"https://www.wikidata.org/wiki/Q722659","display_name":"Graph property","level":5,"score":0.28859999775886536}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.24754","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24754","pdf_url":"https://arxiv.org/pdf/2608.24754","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.2608.24754","doi":"https://doi.org/10.48550/arxiv.2608.24754","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.24754","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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.24754","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24754","pdf_url":"https://arxiv.org/pdf/2608.24754","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":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G7920131011","display_name":null,"funder_award_id":"12431004","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204226322.pdf","grobid_xml":"https://content.openalex.org/works/W7204226322.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Hypercube":[0],"graphs":[1,139],"are":[2,67,75],"fundamental":[3],"model":[4],"spaces":[5],"of":[6,49,144],"positive":[7],"curvature":[8,24],"in":[9,111],"discrete":[10],"comparison":[11],"geometry.":[12],"Let":[13],"$G$":[14,31,112,126],"be":[15,84],"a":[16,46,61,105,123,128],"finite,":[17],"connected,":[18],"simple,":[19],"unweighted":[20],"graph":[21,59,89],"with":[22,72,77,140],"Bakry--\u00c9mery":[23],"bounded":[25],"below":[26],"by":[27],"$K$.":[28,103],"We":[29,41],"call":[30],"Lichnerowicz-sharp":[32,138],"if":[33,107],"its":[34],"first":[35,98],"non-zero":[36],"non-normalized":[37],"Laplacian":[38],"eigenvalue":[39,99],"$\u03bb_1=K$.":[40],"prove":[42],"that,":[43],"after":[44],"removing":[45],"canonical":[47,62,117,142],"collection":[48],"edges":[50],"on":[51],"which":[52],"every":[53,120,133],"$K$-eigenfunction":[54],"is":[55,90,113,122,127],"constant,":[56],"the":[57,87,108],"resulting":[58],"has":[60,97],"bundle":[63],"structure.":[64],"Its":[65],"fibers":[66,143],"regular,":[68],"have":[69],"similar":[70],"structure":[71],"hypercubes,":[73,78],"and":[74,96,125],"Laplacian-cospectral":[76],"although":[79],"they":[80],"need":[81],"not":[82],"themselves":[83],"hypercubes.":[85],"If":[86],"base":[88],"nontrivial,":[91],"then":[92,119],"it":[93],"satisfies":[94],"$\\mathrm{CD}(K,\\infty)$":[95],"strictly":[100],"greater":[101],"than":[102],"As":[104],"consequence,":[106],"vertex":[109],"degree":[110,145],"constant":[114],"along":[115],"each":[116],"fiber,":[118],"fiber":[121],"hypercube":[124,129],"bundle.":[130],"Conversely,":[131],"for":[132],"$d\\geq":[134],"4$,":[135],"we":[136],"construct":[137],"non-hypercube":[141],"$d$.":[146]},"counts_by_year":[],"updated_date":"2026-10-12T09:40:04.860316","created_date":"2026-08-27T00:00:00"}