{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T21:02:00Z","timestamp":1760821320543,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,30]],"date-time":"2022-05-30T00:00:00Z","timestamp":1653868800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["RSP-2021\/401"],"award-info":[{"award-number":["RSP-2021\/401"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A complex fuzzy set (CFS) is described by a complex-valued truth membership function, which is a combination of a standard true membership function plus a phase term. In this paper, we extend the idea of a fuzzy graph (FG) to a complex fuzzy graph (CFG). The CFS complexity arises from the variety of values that its membership function can attain. In contrast to a standard fuzzy membership function, its range is expanded to the complex plane\u2019s unit circle rather than [0,1]. As a result, the CFS provides a mathematical structure for representing membership in a set in terms of complex numbers. In recent times, a mathematical technique has been a popular way to combine several features. Using the preceding mathematical technique, we introduce strong approaches that are properties of CFG. We define the order and size of CFG. We discuss the degree of vertex and the total degree of vertex of CFG. We describe basic operations, including union, join, and the complement of CFG. We show new maximal product and symmetric difference operations on CFG, along with examples and theorems that go along with them. Lastly, at the base of a complex fuzzy graph, we show the application that would be important for measuring the symmetry or asymmetry of acquaintanceship levels of social disease: COVID-19.<\/jats:p>","DOI":"10.3390\/sym14061126","type":"journal-article","created":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T02:30:06Z","timestamp":1653964206000},"page":"1126","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Maximal Product and Symmetric Difference of Complex Fuzzy Graph with Application"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8715-0738","authenticated-orcid":false,"given":"Muhammad","family":"Shoaib","sequence":"first","affiliation":[{"name":"Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan"}]},{"given":"Waqas","family":"Mahmood","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6178-8538","authenticated-orcid":false,"given":"Qin","family":"Xin","sequence":"additional","affiliation":[{"name":"Faculty of Sciences and Technology, University of the Faroe Islands, Vestarabryggja 15, FO 100 Torshavn, Faroe Islands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7855-508X","authenticated-orcid":false,"given":"Fairouz","family":"Tchier","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy Sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1016\/0022-247X(71)90199-5","article-title":"Fuzzy groups","volume":"35","author":"Rosenfeld","year":"1971","journal-title":"J. Math. Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2043","DOI":"10.3233\/JIFS-16433","article-title":"Faces and dual of m-polar fuzzy planar graphs","volume":"31","author":"Ghorai","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_4","first-page":"294","article-title":"On degrees of m-polar fuzzy graph with application","volume":"11","author":"Ghorai","year":"2017","journal-title":"J. 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Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"163260","DOI":"10.1109\/ACCESS.2020.3020366","article-title":"A Certain Class of t-intuitionistic Fuzzy Subgroup","volume":"8","author":"Gulzar","year":"2020","journal-title":"IEEE Access"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"196075","DOI":"10.1109\/ACCESS.2020.3034626","article-title":"A novel applicatioin of complex intutionistic fuzzy sets in group theory","volume":"8","author":"Gulzar","year":"2020","journal-title":"IEEE Access"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"174094","DOI":"10.1109\/ACCESS.2020.3025974","article-title":"Complex Vague Graphs and Their Application in Decision-Making Problems","volume":"8","author":"Zeng","year":"2020","journal-title":"IEEE Access"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1126\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:21:51Z","timestamp":1760138511000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1126"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,30]]},"references-count":22,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061126"],"URL":"https:\/\/doi.org\/10.3390\/sym14061126","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,5,30]]}}}