{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:38:01Z","timestamp":1760236681769,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,12]],"date-time":"2021-12-12T00:00:00Z","timestamp":1639267200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fuzzy graphs (FGs) can play a useful role in natural and human-made structures, including process dynamics in physical, biological, and social systems. Since issues in everyday life are often uncertain due to inconsistent and ambiguous information, it is extremely difficult for an expert to model those difficulties using an FG. Indeterminate and inconsistent information related to real-valued problems can be studied through a picture of the fuzzy graph (PFG), while the FG does not provide mathematically acceptable information. In this regard, we are interested in reducing the limitations of FGs by introducing some new definitions and results for the PFG. This paper aims to describe and explore a few properties of PFGs, including the maximal product (MP), symmetric difference (SD), rejection (RJ), and residue product (RP). Furthermore, we also discuss the degree and total degree of nodes in a PFG. This study also demonstrates the application of a PFG in digital marketing and social networking.<\/jats:p>","DOI":"10.3390\/sym13122400","type":"journal-article","created":{"date-parts":[[2021,12,13]],"date-time":"2021-12-13T01:29:33Z","timestamp":1639358973000},"page":"2400","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Certain Operations on Picture Fuzzy Graph with Application"],"prefix":"10.3390","volume":"13","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, Denmark"}]},{"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":[[2021,12,12]]},"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","unstructured":"Zadeh, L.A., Fu, K.S., and Shimura, M. (2006). Fuzzy graphs. Fuzzy Sets and Their Application, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/0167-8655(87)90012-2","article-title":"Some remarks on fuzzy graphs","volume":"6","author":"Bhattacharya","year":"1987","journal-title":"Pattern Recognit. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/0167-8655(89)90049-4","article-title":"On automorphisms of fuzzy graphs","volume":"9","author":"Bhutani","year":"1989","journal-title":"Pattern Recognit. Lett."},{"key":"ref_5","first-page":"517","article-title":"On Irregular Fuzzy Graphs","volume":"6","author":"Gani","year":"2012","journal-title":"Appl. Math. Sci."},{"key":"ref_6","first-page":"145","article-title":"Order and Size in Fuzzy Graphs","volume":"22","author":"Gani","year":"2003","journal-title":"Bull. Pure Appl. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/0020-0255(94)90116-3","article-title":"Operations on fuzzy graphs","volume":"79","author":"Mordeson","year":"1994","journal-title":"Inform. Sci."},{"key":"ref_8","unstructured":"Mathew, S., and Sunitha, M.S. (2012). Fuzzy Graphs, Basic Concepts and Applications, Lap Lambert Academic Publishing."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Shao, Z., Kosari, S., Rashmanlou, H., and Shoaib, M. (2020). New Concepts in Intuitionistic Fuzzy Graph with Application in Water Supplier Systems. Mathematics, 8.","DOI":"10.3390\/math8081241"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1080\/18756891.2015.1063243","article-title":"Bipolar fuzzy graphs with categorical properties","volume":"8","author":"Rashmanlou","year":"2015","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"571","DOI":"10.3233\/IFS-141333","article-title":"A study on bipolar fuzzy graphs","volume":"28","author":"Rashmanlou","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/03081079.2015.1072521","article-title":"Product of bipolar fuzzy graphs and their degree","volume":"45","author":"Rashmanlou","year":"2016","journal-title":"Int. J. Gen. Syst."},{"key":"ref_13","first-page":"1756","article-title":"Some properties of highly irregular interval-valued fuzzy graphs","volume":"27","author":"Rashmanlou","year":"2013","journal-title":"World Appl. Sci. J."},{"key":"ref_14","first-page":"43","article-title":"Balanced interval-valued fuzzy graph","volume":"17","author":"Rashmanlou","year":"2013","journal-title":"J. Phys. Sci."},{"key":"ref_15","first-page":"107","article-title":"Antipodal interval-valued fuzzy graphs","volume":"3","author":"Rashmanlou","year":"2013","journal-title":"Int. J. Appl. Fuzzy Sets Articial Intell."},{"key":"ref_16","first-page":"677","article-title":"Complete interval-valued fuzzy graphs","volume":"6","author":"Rashmanlou","year":"2013","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_17","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_18","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_19","first-page":"5253346","article-title":"On the Algeberaic Characteristics of Fuzzy Sub-e-group","volume":"2021","author":"Bhunia","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_20","first-page":"287","article-title":"Neutrosophic set a generalization of intuitionistic fuzzy sets","volume":"24","author":"Smarandache","year":"2015","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_21","first-page":"547","article-title":"The classes of bipolar single-valued neutrosophic graphs","volume":"10","author":"Hassan","year":"2020","journal-title":"TWMS J. Appl. Eng. Math."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Zuo, C., Pal, A., and Dey, A. (2019). New Concepts of Picture Fuzzy Graphs with Application. Mathematics, 7.","DOI":"10.3390\/math7050470"},{"key":"ref_23","first-page":"409","article-title":"Picture fuzzy sets","volume":"30","author":"Cuong","year":"2014","journal-title":"J. Comput. Sci. Cybern."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Wang, R., Wang, J., Gao, H., and Wei, G. (2019). Methods for MADM with picture fuzzy muirhead mean operatorsand their application for evaluating the financial investment risk. Symmetry, 11.","DOI":"10.3390\/sym11010006"},{"key":"ref_25","first-page":"77","article-title":"Some similarity measures for picture fuzzy sets and their applications","volume":"15","author":"Wei","year":"2018","journal-title":"Iran. J. Fuzzy Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2400\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:46:02Z","timestamp":1760168762000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2400"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,12]]},"references-count":25,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["sym13122400"],"URL":"https:\/\/doi.org\/10.3390\/sym13122400","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,12,12]]}}}