{"id":"https://openalex.org/W7130859646","doi":"https://doi.org/10.71602/tfss.2026.1217721","title":"Closeness Centrality of Cycle and Corona Product Graphs: Application in Fuzzy Transportation Network","display_name":"Closeness Centrality of Cycle and Corona Product Graphs: Application in Fuzzy Transportation Network","publication_year":2025,"publication_date":"2025-09-12","ids":{"openalex":"https://openalex.org/W7130859646","doi":"https://doi.org/10.71602/tfss.2026.1217721"},"language":"en","primary_location":{"id":"pmh:doi:10.71602/tfss.2026.1217721","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"JournalArticle"},"type":"article","indexed_in":["datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5126556670","display_name":"Shaoli Nandi","orcid":null},"institutions":[{"id":"https://openalex.org/I169046204","display_name":"Universidad Aut\u00f3noma de Nuevo Le\u00f3n","ror":"https://ror.org/01fh86n78","country_code":"MX","type":"education","lineage":["https://openalex.org/I169046204"]}],"countries":["MX"],"is_corresponding":true,"raw_author_name":"Nandi, Shaoli","raw_affiliation_strings":["Department of Mathematics, Raja N. L. Khan Womens College (Autonomous), Midnapore, India"],"raw_orcid":"https://orcid.org/0000-0002-0529-922X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Raja N. L. Khan Womens College (Autonomous), Midnapore, India","institution_ids":["https://openalex.org/I169046204"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087598067","display_name":"Sukumar Mondal","orcid":"https://orcid.org/0000-0002-5639-8439"},"institutions":[{"id":"https://openalex.org/I169046204","display_name":"Universidad Aut\u00f3noma de Nuevo Le\u00f3n","ror":"https://ror.org/01fh86n78","country_code":"MX","type":"education","lineage":["https://openalex.org/I169046204"]}],"countries":["MX"],"is_corresponding":false,"raw_author_name":"Mondal, Sukumar","raw_affiliation_strings":["Department of Mathematics, Raja N. L. Khan Womens College (Autonomous), Midnapore, India"],"raw_orcid":"https://orcid.org/0000-0002-5639-8439","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Raja N. L. Khan Womens College (Autonomous), Midnapore, India","institution_ids":["https://openalex.org/I169046204"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5058160355","display_name":"Sambhu Charan Barman","orcid":"https://orcid.org/0000-0002-6662-5172"},"institutions":[{"id":"https://openalex.org/I4210108373","display_name":"Ashwini Hospital","ror":"https://ror.org/01hcfbh13","country_code":"IN","type":"healthcare","lineage":["https://openalex.org/I4210108373"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Barman, Sambhu Charan","raw_affiliation_strings":["Department of Mathematics, Shahid Matangini Hazra Government General Degree College for Women, Medinipur, India"],"raw_orcid":"https://orcid.org/0000-0002-6662-5172","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shahid Matangini Hazra Government General Degree College for Women, Medinipur, India","institution_ids":["https://openalex.org/I4210108373"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5126556670"],"corresponding_institution_ids":["https://openalex.org/I169046204"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.71978721,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"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/T10064","display_name":"Complex Network Analysis Techniques","score":0.8865000009536743,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.8865000009536743,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.0771000012755394,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.008999999612569809,"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/centrality","display_name":"Centrality","score":0.958899974822998},{"id":"https://openalex.org/keywords/closeness","display_name":"Closeness","score":0.7003999948501587},{"id":"https://openalex.org/keywords/betweenness-centrality","display_name":"Betweenness centrality","score":0.6205000281333923},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.5430999994277954},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5113000273704529},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.4657999873161316},{"id":"https://openalex.org/keywords/network-science","display_name":"Network science","score":0.4325000047683716},{"id":"https://openalex.org/keywords/node","display_name":"Node (physics)","score":0.39399999380111694}],"concepts":[{"id":"https://openalex.org/C53811970","wikidata":"https://www.wikidata.org/wiki/Q5062194","display_name":"Centrality","level":2,"score":0.958899974822998},{"id":"https://openalex.org/C2779545769","wikidata":"https://www.wikidata.org/wiki/Q5135364","display_name":"Closeness","level":2,"score":0.7003999948501587},{"id":"https://openalex.org/C117045392","wikidata":"https://www.wikidata.org/wiki/Q4899215","display_name":"Betweenness centrality","level":3,"score":0.6205000281333923},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5430999994277954},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5299999713897705},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5113000273704529},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.4657999873161316},{"id":"https://openalex.org/C137753397","wikidata":"https://www.wikidata.org/wiki/Q2434424","display_name":"Network science","level":3,"score":0.4325000047683716},{"id":"https://openalex.org/C62611344","wikidata":"https://www.wikidata.org/wiki/Q1062658","display_name":"Node (physics)","level":2,"score":0.39399999380111694},{"id":"https://openalex.org/C11804174","wikidata":"https://www.wikidata.org/wiki/Q1192971","display_name":"Network theory","level":2,"score":0.3871999979019165},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.38510000705718994},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3538999855518341},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.3483000099658966},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.30880001187324524},{"id":"https://openalex.org/C131349576","wikidata":"https://www.wikidata.org/wiki/Q6378571","display_name":"Katz centrality","level":4,"score":0.30649998784065247},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.3059999942779541},{"id":"https://openalex.org/C82927061","wikidata":"https://www.wikidata.org/wiki/Q178192","display_name":"Cross product","level":2,"score":0.305400013923645},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.298799991607666},{"id":"https://openalex.org/C32946077","wikidata":"https://www.wikidata.org/wiki/Q618079","display_name":"Network analysis","level":2,"score":0.2953999936580658},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.2815999984741211},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.2603999972343445},{"id":"https://openalex.org/C34947359","wikidata":"https://www.wikidata.org/wiki/Q665189","display_name":"Complex network","level":2,"score":0.25940001010894775},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.2556999921798706}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:doi:10.71602/tfss.2026.1217721","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"JournalArticle"},{"id":"pmh:oai:doaj.org/article:c78301de3091418c85717cf40ec92fae","is_oa":true,"landing_page_url":"https://doaj.org/article/c78301de3091418c85717cf40ec92fae","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Transactions on Fuzzy Sets and Systems, Vol 5, Iss 1, Pp 155-181 (2026)","raw_type":"article"},{"id":"doi:10.71602/tfss.2026.1217721","is_oa":true,"landing_page_url":"https://doi.org/10.71602/tfss.2026.1217721","pdf_url":null,"source":{"id":"https://openalex.org/S169411308","display_name":"Fuzzy Sets and Systems","issn_l":"0165-0114","issn":["0165-0114","1872-6801"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:doi:10.71602/tfss.2026.1217721","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"JournalArticle"},"sustainable_development_goals":[{"display_name":"Decent work and economic growth","score":0.4097585678100586,"id":"https://metadata.un.org/sdg/8"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Centrality":[0],"measurement":[1,95],"is":[2,60,96],"a":[3,17,54,58,83,190,225],"useful":[4],"way":[5],"to":[6,30,72,233],"find":[7],"the":[8,61,65,68,155,164,169,210,218,239],"most":[9],"important":[10],"points":[11],"(vertices)":[12],"and":[13,32,149,208,237],"connections":[14],"(edges)":[15],"in":[16,180,201,224],"network\u200e.":[18],"\u200eOver":[19],"time\u200e,":[20],"\u200emany":[21],"researchers":[22],"have":[23],"created":[24],"different":[25],"types":[26],"of":[27,53,57,64,67,82,125,168,193,220],"centrality":[28,52,80,94,101,124,167],"measures":[29],"study":[31],"understand":[33],"how":[34],"networks":[35,203],"work\u200e.":[36],"\u200eCloseness":[37],"centrality\u200e,":[38],"\u200ein":[39],"particular\u200e,":[40],"\u200eis":[41],"crucial":[42],"for":[43,120,197],"examining":[44],"biological\u200e,":[45],"\u200esocial\u200e,":[46],"\u200eand":[47,185],"transportation":[48,226],"networks\u200e.":[49,187],"\u200eThe":[50,173],"closeness":[51,79,123,166],"node":[55],"$u$":[56,71,85],"graph":[59,171,175,213,242],"multiplicative":[62],"inverse":[63],"sum":[66],"distances":[69],"from":[70],"each":[73],"other":[74],"vertex\u200e.":[75],"\u200eWe":[76,152,188,215],"define":[77],"normalized":[78,122],"$C_{NC}(u)$":[81],"vertex":[84,165],"as":[86,106,108],"$C_{NC}(u)":[87],"=":[88],"\\frac{n-1}{\\sum_{x\\in":[89],"V}\\mathfrak{d}(u,x)}$\u200e,":[90],"\u200ewhere":[91],"$n=|V|$\u200e.":[92],"\u200eThis":[93],"more":[97],"receivable":[98],"than":[99],"degree":[100,236],"because":[102],"it":[103],"counts":[104],"direct":[105],"well":[107],"indirect":[109],"connections\u200e.":[110],"\u200eIn":[111],"this":[112],"paper\u200e,":[113],"\u200ewe":[114],"present":[115,217],"some":[116,126],"new":[117],"theoretical":[118],"results":[119,196,207,223],"finding":[121,163],"corona":[127,174,211,240],"product":[128,212,241],"graphs":[129],"like":[130],"$P_{n}\\odot":[131],"P_{m}$\u200e,":[132,142],"\u200e$P_{n}\\odot":[133,137,143],"K_{m}$\u200e,":[134,136,146],"\u200e$C_{n}\\odot":[135,139,141],"C_{m}$\u200e,":[138,140],"S_{m}$\u200e,":[144],"\u200e$S_{n}\\odot":[145],"\u200e$K_{n}\\odot":[147],"P_{m}$":[148],"$K_{n}\\odot":[150],"K_{m}$\u200e.":[151],"also":[153,216],"correct":[154],"result":[156,232],"established":[157],"by":[158,228],"Eballe":[159],"et":[160],"al\u200e.":[161],"\u200efor":[162],"cycle":[170],"$C_{n}$\u200e.":[172],"has":[176],"many":[177],"applications\u200e,":[178],"\u200eincluding":[179],"signed":[181],"networks\u200e,":[182],"\u200ebiotechnology\u200e,":[183],"\u200echemistry\u200e,":[184],"small-world":[186,202],"demonstrate":[189],"practical":[191],"application":[192],"our":[194,206,221,230],"proposed":[195],"identifying":[198],"influential":[199],"nodes":[200],"based":[204],"on":[205],"using":[209,238],"model\u200e.":[214,243],"applications":[219],"studied":[222],"network":[227],"transferring":[229],"crisp":[231],"fuzzy":[234],"membership":[235]},"counts_by_year":[],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2026-02-22T00:00:00"}