{"id":"https://openalex.org/W4281976241","doi":"https://doi.org/10.3390/sym14061126","title":"Maximal Product and Symmetric Difference of Complex Fuzzy Graph with Application","display_name":"Maximal Product and Symmetric Difference of Complex Fuzzy Graph with Application","publication_year":2022,"publication_date":"2022-05-30","ids":{"openalex":"https://openalex.org/W4281976241","doi":"https://doi.org/10.3390/sym14061126"},"language":"en","primary_location":{"id":"doi:10.3390/sym14061126","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061126","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1126/pdf?version=1654076943","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/6/1126/pdf?version=1654076943","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101518078","display_name":"Muhammad Shoaib","orcid":"https://orcid.org/0000-0001-8715-0738"},"institutions":[{"id":"https://openalex.org/I12469534","display_name":"Quaid-i-Azam University","ror":"https://ror.org/04s9hft57","country_code":"PK","type":"education","lineage":["https://openalex.org/I12469534"]}],"countries":["PK"],"is_corresponding":true,"raw_author_name":"Muhammad Shoaib","raw_affiliation_strings":["Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan"],"raw_orcid":"https://orcid.org/0000-0001-8715-0738","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan","institution_ids":["https://openalex.org/I12469534"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5051931629","display_name":"Waqas Mahmood","orcid":"https://orcid.org/0000-0003-1171-8782"},"institutions":[{"id":"https://openalex.org/I12469534","display_name":"Quaid-i-Azam University","ror":"https://ror.org/04s9hft57","country_code":"PK","type":"education","lineage":["https://openalex.org/I12469534"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Waqas Mahmood","raw_affiliation_strings":["Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan","institution_ids":["https://openalex.org/I12469534"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050608316","display_name":"Qin Xin","orcid":"https://orcid.org/0000-0002-6178-8538"},"institutions":[{"id":"https://openalex.org/I103146751","display_name":"University of the Faroe Islands","ror":"https://ror.org/05mwmd090","country_code":"FO","type":"education","lineage":["https://openalex.org/I103146751"]}],"countries":["FO"],"is_corresponding":false,"raw_author_name":"Qin Xin","raw_affiliation_strings":["Faculty of Sciences and Technology, University of the Faroe Islands, Vestarabryggja 15, FO 100 Torshavn, Faroe Islands"],"raw_orcid":"https://orcid.org/0000-0002-6178-8538","affiliations":[{"raw_affiliation_string":"Faculty of Sciences and Technology, University of the Faroe Islands, Vestarabryggja 15, FO 100 Torshavn, Faroe Islands","institution_ids":["https://openalex.org/I103146751"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5087794334","display_name":"Fairouz Tchier","orcid":"https://orcid.org/0000-0001-7855-508X"},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Fairouz Tchier","raw_affiliation_strings":["Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-7855-508X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5101518078"],"corresponding_institution_ids":["https://openalex.org/I12469534"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.1821,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.80195513,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"14","issue":"6","first_page":"1126","last_page":"1126"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11671","display_name":"Fuzzy and Soft Set Theory","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9837999939918518,"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/mathematics","display_name":"Mathematics","score":0.6276777982711792},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6032184958457947},{"id":"https://openalex.org/keywords/membership-function","display_name":"Membership function","score":0.5685341954231262},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.5039142966270447},{"id":"https://openalex.org/keywords/fuzzy-number","display_name":"Fuzzy number","score":0.4836210310459137},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.4533795416355133},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4449392259120941},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4274926483631134},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.35413485765457153},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2644670605659485},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.166225403547287}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6276777982711792},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6032184958457947},{"id":"https://openalex.org/C5263885","wikidata":"https://www.wikidata.org/wiki/Q1967302","display_name":"Membership function","level":4,"score":0.5685341954231262},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.5039142966270447},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.4836210310459137},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.4533795416355133},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4449392259120941},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4274926483631134},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.35413485765457153},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2644670605659485},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.166225403547287}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14061126","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061126","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1126/pdf?version=1654076943","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:60b7ae9bb7064d4a92e43ab46d8379f6","is_oa":true,"landing_page_url":"https://doaj.org/article/60b7ae9bb7064d4a92e43ab46d8379f6","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":"Symmetry, Vol 14, Iss 6, p 1126 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/6/1126/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14061126","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14061126","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061126","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1126/pdf?version=1654076943","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.7300000190734863}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4281976241.pdf"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W1601092472","https://openalex.org/W1980564456","https://openalex.org/W2011023980","https://openalex.org/W2028649872","https://openalex.org/W2097403045","https://openalex.org/W2184523047","https://openalex.org/W2291770014","https://openalex.org/W2490481354","https://openalex.org/W3045598505","https://openalex.org/W3081525434","https://openalex.org/W3087798822","https://openalex.org/W3095983688","https://openalex.org/W3096389194","https://openalex.org/W3103076968","https://openalex.org/W3159839367","https://openalex.org/W4200075406","https://openalex.org/W4210778630","https://openalex.org/W4211007335","https://openalex.org/W6680390941","https://openalex.org/W6686010921","https://openalex.org/W6784636093","https://openalex.org/W6785312522"],"related_works":["https://openalex.org/W4236695056","https://openalex.org/W87675884","https://openalex.org/W154406135","https://openalex.org/W2804121587","https://openalex.org/W3186561448","https://openalex.org/W2132859092","https://openalex.org/W2116249946","https://openalex.org/W1486684451","https://openalex.org/W4361273540","https://openalex.org/W4361984403"],"abstract_inverted_index":{"A":[0],"complex":[1,41,74,99,191],"fuzzy":[2,36,42,65,192],"set":[3,95],"(CFS)":[4],"is":[5,14,70],"described":[6],"by":[7],"a":[8,15,18,24,35,40,63,82,87,94,104,109,190],"complex-valued":[9],"truth":[10],"membership":[11,21,56,66,92],"function,":[12,67],"which":[13],"combination":[16],"of":[17,34,52,98,128,136,142,148,150,162,189,208,211],"standard":[19,64],"true":[20],"function":[22,57],"plus":[23],"phase":[25],"term.":[26],"In":[27,60,101],"this":[28],"paper,":[29],"we":[30,121,194],"extend":[31],"the":[32,50,73,84,117,132,140,145,160,187,196,204],"idea":[33],"graph":[37,43],"(FG)":[38],"to":[39,62,72,112],"(CFG).":[44],"The":[45],"CFS":[46,85],"complexity":[47],"arises":[48],"from":[49],"variety":[51],"values":[53],"that":[54,125,180,198],"its":[55,68],"can":[58],"attain.":[59],"contrast":[61],"range":[69],"expanded":[71],"plane\u2019s":[75],"unit":[76],"circle":[77],"rather":[78],"than":[79],"[0,1].":[80],"As":[81],"result,":[83],"provides":[86],"mathematical":[88,105,119],"structure":[89],"for":[90,202],"representing":[91],"in":[93,96],"terms":[97],"numbers.":[100],"recent":[102],"times,":[103],"technique":[106],"has":[107],"been":[108],"popular":[110],"way":[111],"combine":[113],"several":[114],"features.":[115],"Using":[116],"preceding":[118],"technique,":[120],"introduce":[122],"strong":[123],"approaches":[124],"are":[126],"properties":[127],"CFG.":[129,137,151,163],"We":[130,138,152,164],"define":[131],"order":[133],"and":[134,144,159,169,178],"size":[135],"discuss":[139],"degree":[141,147],"vertex":[143,149],"total":[146],"describe":[153],"basic":[154],"operations,":[155],"including":[156],"union,":[157],"join,":[158],"complement":[161],"show":[165,195],"new":[166],"maximal":[167],"product":[168],"symmetric":[170],"difference":[171],"operations":[172],"on":[173],"CFG,":[174],"along":[175,182],"with":[176,183],"examples":[177],"theorems":[179],"go":[181],"them.":[184],"Lastly,":[185],"at":[186],"base":[188],"graph,":[193],"application":[197],"would":[199],"be":[200],"important":[201],"measuring":[203],"symmetry":[205],"or":[206],"asymmetry":[207],"acquaintanceship":[209],"levels":[210],"social":[212],"disease:":[213],"COVID-19.":[214]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":1}],"updated_date":"2026-05-22T06:13:13.366637","created_date":"2025-10-10T00:00:00"}
