{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,19]],"date-time":"2025-08-19T10:58:16Z","timestamp":1755601096087},"reference-count":14,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Computational Intelligence and Soft Computing"],"published-print":{"date-parts":[[2016]]},"abstract":"<jats:p>A graph structure is a useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. In this paper, we apply the concept of bipolar fuzzy sets to graph structures. We introduce certain notions, including bipolar fuzzy graph structure (BFGS), strong bipolar fuzzy graph structure, bipolar fuzzy<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>i<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-cycle, bipolar fuzzy<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>i<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-tree, bipolar fuzzy<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>i<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-cut vertex, and bipolar fuzzy<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>i<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-bridge, and illustrate these notions by several examples. We study<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>\u03d5<\/mml:mi><\/mml:mrow><\/mml:math>-complement, self-complement, strong self-complement, and totally strong self-complement in bipolar fuzzy graph structures, and we investigate some of their interesting properties.<\/jats:p>","DOI":"10.1155\/2016\/5859080","type":"journal-article","created":{"date-parts":[[2016,2,10]],"date-time":"2016-02-10T21:05:11Z","timestamp":1455138311000},"page":"1-13","source":"Crossref","is-referenced-by-count":25,"title":["Application of Bipolar Fuzzy Sets in Graph Structures"],"prefix":"10.1155","volume":"2016","author":[{"given":"Muhammad","family":"Akram","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Pakistan"}]},{"given":"Rabia","family":"Akmal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Pakistan"}]}],"member":"98","reference":[{"issue":"2","key":"15","first-page":"65","volume":"3","year":"2006","journal-title":"Bulletin of Kerala Mathematics Association"},{"key":"17","first-page":"338","volume":"8","year":"1965","journal-title":"Information and Computation"},{"key":"10","volume":"1","year":"1973"},{"key":"14","first-page":"77","volume-title":"Fuzzy graphs","year":"1975"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/0167-8655(87)90012-2"},{"key":"12","series-title":"Second Edition 2001","year":"1998"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2011.07.037"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.knosys.2012.08.022"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1007\/s00521-011-0772-6"},{"key":"4","first-page":"97","volume":"31","year":"2013","journal-title":"Italian Journal of Pure and Applied Mathematics"},{"key":"5","first-page":"426","volume":"31","year":"2013","journal-title":"Italian Journal of Pure and Applied Mathematics"},{"key":"6","first-page":"385","volume":"121","year":"2015","journal-title":"Ars Combinatoria"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2009.10.006"},{"issue":"2","key":"18","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0020-0255(71)80005-1","volume":"3","year":"1971","journal-title":"Information Sciences"}],"container-title":["Applied Computational Intelligence and Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/acisc\/2016\/5859080.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/acisc\/2016\/5859080.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/acisc\/2016\/5859080.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,24]],"date-time":"2017-06-24T05:38:35Z","timestamp":1498282715000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/acisc\/2016\/5859080\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"references-count":14,"alternative-id":["5859080","5859080"],"URL":"https:\/\/doi.org\/10.1155\/2016\/5859080","relation":{},"ISSN":["1687-9724","1687-9732"],"issn-type":[{"value":"1687-9724","type":"print"},{"value":"1687-9732","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016]]}}}