{"id":"https://openalex.org/W1541802130","doi":"https://doi.org/10.37236/150","title":"Multipartite Separability of Laplacian Matrices of Graphs","display_name":"Multipartite Separability of Laplacian Matrices of Graphs","publication_year":2009,"publication_date":"2009-05-15","ids":{"openalex":"https://openalex.org/W1541802130","doi":"https://doi.org/10.37236/150","mag":"1541802130"},"language":"en","primary_location":{"id":"doi:10.37236/150","is_oa":true,"landing_page_url":"https://doi.org/10.37236/150","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r61/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r61/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038483763","display_name":"Chai Wah Wu","orcid":"https://orcid.org/0000-0002-0657-0683"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Chai Wah Wu","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5038483763"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3875,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.53271115,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"16","issue":"1","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":0.9993000030517578,"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"}},"topics":[{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9993000030517578,"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"}},{"id":"https://openalex.org/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T10020","display_name":"Quantum Information and Cryptography","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/multipartite","display_name":"Multipartite","score":0.8777238130569458},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7391179800033569},{"id":"https://openalex.org/keywords/bipartite-graph","display_name":"Bipartite graph","score":0.6920928955078125},{"id":"https://openalex.org/keywords/laplacian-matrix","display_name":"Laplacian matrix","score":0.6546202898025513},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.61866295337677},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.5502467751502991},{"id":"https://openalex.org/keywords/laplace-operator","display_name":"Laplace operator","score":0.528663158416748},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.46476373076438904},{"id":"https://openalex.org/keywords/separable-space","display_name":"Separable space","score":0.45804116129875183},{"id":"https://openalex.org/keywords/cograph","display_name":"Cograph","score":0.4386158883571625},{"id":"https://openalex.org/keywords/quantum","display_name":"Quantum","score":0.33893582224845886},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.3185170292854309},{"id":"https://openalex.org/keywords/1-planar-graph","display_name":"1-planar graph","score":0.27151045203208923},{"id":"https://openalex.org/keywords/chordal-graph","display_name":"Chordal graph","score":0.2246887981891632},{"id":"https://openalex.org/keywords/quantum-entanglement","display_name":"Quantum entanglement","score":0.17075243592262268},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08947890996932983},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.062418580055236816}],"concepts":[{"id":"https://openalex.org/C166733315","wikidata":"https://www.wikidata.org/wiki/Q6934831","display_name":"Multipartite","level":4,"score":0.8777238130569458},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7391179800033569},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.6920928955078125},{"id":"https://openalex.org/C115178988","wikidata":"https://www.wikidata.org/wiki/Q772067","display_name":"Laplacian matrix","level":3,"score":0.6546202898025513},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.61866295337677},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5502467751502991},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.528663158416748},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46476373076438904},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.45804116129875183},{"id":"https://openalex.org/C59824394","wikidata":"https://www.wikidata.org/wiki/Q5141281","display_name":"Cograph","level":5,"score":0.4386158883571625},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.33893582224845886},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3185170292854309},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.27151045203208923},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.2246887981891632},{"id":"https://openalex.org/C121040770","wikidata":"https://www.wikidata.org/wiki/Q215675","display_name":"Quantum entanglement","level":3,"score":0.17075243592262268},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08947890996932983},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.062418580055236816},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/150","is_oa":true,"landing_page_url":"https://doi.org/10.37236/150","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r61/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/150","is_oa":true,"landing_page_url":"https://doi.org/10.37236/150","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r61/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1541802130.pdf","grobid_xml":"https://content.openalex.org/works/W1541802130.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W1492379432","https://openalex.org/W1631356911","https://openalex.org/W2058988058","https://openalex.org/W2140451017","https://openalex.org/W2764347725","https://openalex.org/W3100495106","https://openalex.org/W4238591275","https://openalex.org/W6629125196"],"related_works":["https://openalex.org/W2383428243","https://openalex.org/W1954384522","https://openalex.org/W2167462019","https://openalex.org/W4390722723","https://openalex.org/W2730710490","https://openalex.org/W2051247518","https://openalex.org/W4284883459","https://openalex.org/W4380714634","https://openalex.org/W2033371793","https://openalex.org/W4288409989"],"abstract_inverted_index":{"Recently,":[0],"Braunstein":[1,94],"et":[2,95],"al.":[3,96],"introduced":[4],"normalized":[5,53,81],"Laplacian":[6,41,54,82,102],"matrices":[7,12,42],"of":[8,28,40,43,104,107],"graphs":[9,51,108],"as":[10],"density":[11],"in":[13,93],"quantum":[14,21],"mechanics":[15],"and":[16,24,122],"studied":[17],"the":[18,29,37,68,80,101],"relationships":[19],"between":[20],"physical":[22],"properties":[23,27],"graph":[25],"theoretical":[26],"underlying":[30],"graphs.":[31,44],"We":[32],"provide":[33],"further":[34],"results":[35,87,119],"on":[36,67],"multipartite":[38,57,115],"separability":[39],"In":[45],"particular,":[46],"we":[47,64,98],"identify":[48],"complete":[49],"bipartite":[50,121],"whose":[52],"matrix":[55,83,103],"is":[56,74,84,114],"entangled":[58],"under":[59,78],"any":[60,105],"vertex":[61,69,76],"labeling.":[62],"Furthermore,":[63],"give":[65],"conditions":[66],"degrees":[70],"such":[71],"that":[72,100],"there":[73],"a":[75],"labeling":[77],"which":[79],"entangled.":[85],"These":[86],"address":[88],"an":[89],"open":[90],"question":[91],"raised":[92],"Finally,":[97],"show":[99],"product":[106],"(strong,":[109],"Cartesian,":[110],"tensor,":[111],"lexicographical,":[112],"etc.)":[113],"separable,":[116],"extending":[117],"analogous":[118],"for":[120],"tripartite":[123],"separability.":[124]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}