{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T15:13:10Z","timestamp":1746803590685,"version":"3.40.5"},"reference-count":8,"publisher":"Wiley","license":[{"start":{"date-parts":[[2017,1,1]],"date-time":"2017-01-01T00:00:00Z","timestamp":1483228800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Fuzzy Systems"],"published-print":{"date-parts":[[2017]]},"abstract":"<jats:p>Computational intelligence and computer science rely on graph theory to solve combinatorial problems. Normal product and tensor product of an <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:math>-polar fuzzy graph have been introduced in this article. Degrees of vertices in various product graphs, like Cartesian product, composition, tensor product, and normal product, have been computed. Complement and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>\u03bc<\/mml:mi><\/mml:mrow><\/mml:math>-complement of an <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:math>-polar fuzzy graph are defined and some properties are studied. An application of an <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:math>-polar fuzzy graph is also presented in this article.<\/jats:p>","DOI":"10.1155\/2017\/3817469","type":"journal-article","created":{"date-parts":[[2017,8,22]],"date-time":"2017-08-22T17:01:48Z","timestamp":1503421308000},"page":"1-9","source":"Crossref","is-referenced-by-count":5,"title":["Vertex Degrees and Isomorphic Properties in Complement of an <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:math>-Polar Fuzzy Graph"],"prefix":"10.1155","volume":"2017","author":[{"given":"Ch.","family":"Ramprasad","sequence":"first","affiliation":[{"name":"Department of Mathematics, Vasireddy Venkatadri Institute of Technology, Nambur 522 508, India"},{"name":"Department of Mathematics, VFSTR University, Vadlamudi 522 237, India"}]},{"given":"P. L. N.","family":"Varma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, VFSTR University, Vadlamudi 522 237, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8285-113X","authenticated-orcid":true,"given":"S.","family":"Satyanarayana","sequence":"additional","affiliation":[{"name":"Department of CSE, KL University, Vaddeswaram 522502, India"}]},{"given":"N.","family":"Srinivasarao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tirumala Engineering College, Narasaraopet 522034, India"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2011.07.037"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8655(89)90049-4"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1155\/2014\/416530"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.psra.2016.06.004"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0255(94)90116-3"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1080\/18756891.2015.1063243"},{"issue":"9","key":"7","first-page":"1451","volume":"33","year":"2002","journal-title":"Indian Journal of Pure and Applied Mathematics"},{"key":"8","first-page":"264","volume":"73","year":"2017","journal-title":"Ponte"}],"container-title":["Advances in Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/afs\/2017\/3817469.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/afs\/2017\/3817469.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/afs\/2017\/3817469.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,8,22]],"date-time":"2017-08-22T17:01:48Z","timestamp":1503421308000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/afs\/2017\/3817469\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017]]},"references-count":8,"alternative-id":["3817469","3817469"],"URL":"https:\/\/doi.org\/10.1155\/2017\/3817469","relation":{},"ISSN":["1687-7101","1687-711X"],"issn-type":[{"type":"print","value":"1687-7101"},{"type":"electronic","value":"1687-711X"}],"subject":[],"published":{"date-parts":[[2017]]}}}