{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T01:50:54Z","timestamp":1782352254173,"version":"3.54.5"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T00:00:00Z","timestamp":1668556800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T00:00:00Z","timestamp":1668556800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Fundamental Research Program of Shanxi Province","award":["No. 202103021223272"],"award-info":[{"award-number":["No. 202103021223272"]}]},{"name":"Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi","award":["No. 2021L305"],"award-info":[{"award-number":["No. 2021L305"]}]},{"DOI":"10.13039\/501100003398","name":"Shanxi Scholarship Council of China","doi-asserted-by":"publisher","award":["No.2020-122"],"award-info":[{"award-number":["No.2020-122"]}],"id":[{"id":"10.13039\/501100003398","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Research funding of Shanxi Province for outstanding talents","award":["No. 20222003"],"award-info":[{"award-number":["No. 20222003"]}]},{"name":"Taiyuan University of Science and Technology Scientific Research Initial Funding","award":["No. 20202049"],"award-info":[{"award-number":["No. 20202049"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2023,2]]},"DOI":"10.1007\/s00500-022-07636-1","type":"journal-article","created":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T18:03:11Z","timestamp":1668621791000},"page":"1367-1378","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Fuzzy average edge connectivity with its application to communication networks"],"prefix":"10.1007","volume":"27","author":[{"given":"Junye","family":"Ma","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9434-7800","authenticated-orcid":false,"given":"Lin","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jing","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,11,16]]},"reference":[{"key":"7636_CR1","volume-title":"m-Polar fuzzy graphs theory, methods and applications","author":"M Akram","year":"2019","unstructured":"Akram M (2019) m-Polar fuzzy graphs theory, methods and applications. Springer, Cham"},{"issue":"1","key":"7636_CR2","doi-asserted-by":"publisher","first-page":"1553","DOI":"10.3233\/JIFS-210411","volume":"41","author":"M Akram","year":"2021","unstructured":"Akram M, Siddique S, Ahmad U (2021) Menger\u2019s theorem for m-polar fuzzy graphs and application of m-polar fuzzy edges to road network. J Intell Fuzzy Syst 41(1):1553\u20131574","journal-title":"J Intell Fuzzy Syst"},{"issue":"1","key":"7636_CR3","doi-asserted-by":"publisher","first-page":"420","DOI":"10.3934\/mbe.2022021","volume":"19","author":"M Akram","year":"2022","unstructured":"Akram M, Siddique S, Alharbi MG (2022) Clustering algorithm with strength of connectedness for m-polar fuzzy network models. Math Biosci Eng 19(1):420\u2013455","journal-title":"Math Biosci Eng"},{"issue":"4","key":"7636_CR4","doi-asserted-by":"publisher","first-page":"1659","DOI":"10.3233\/IFS-141451","volume":"28","author":"N Anjali","year":"2015","unstructured":"Anjali N, Mathew S (2015) On blocks and stars in fuzzy graphs. J Intell Fuzzy Syst 28(4):1659\u20131665","journal-title":"J Intell Fuzzy Syst"},{"issue":"7","key":"7636_CR5","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1016\/0167-8655(91)90316-E","volume":"12","author":"S Banerjee","year":"1991","unstructured":"Banerjee S (1991) An optimal algorithm to find the degrees of connectedness in an undirected edge-weighted graph. Pattern Recognit Lett 12(7):421\u2013424","journal-title":"Pattern Recognit Lett"},{"issue":"3","key":"7636_CR6","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/0167-8655(89)90049-4","volume":"9","author":"KR Bhutani","year":"1989","unstructured":"Bhutani KR (1989) On automorphisms of fuzzy graphs. Pattern Recognit Lett 9(3):159\u2013162","journal-title":"Pattern Recognit Lett"},{"issue":"1\u20132","key":"7636_CR7","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/S0020-0255(03)00157-9","volume":"155","author":"KR Bhutani","year":"2003","unstructured":"Bhutani KR, Battou A (2003) On m-strong fuzzy graphs. Inf Sci 155(1\u20132):103\u2013109","journal-title":"Inf Sci"},{"key":"7636_CR8","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1016\/S0020-0255(03)00078-1","volume":"152","author":"KR Bhutani","year":"2003","unstructured":"Bhutani KR, Rosenfeld A (2003) Fuzzy end nodes in fuzzy graphs. Inf Sci 152:323\u2013326","journal-title":"Inf Sci"},{"key":"7636_CR9","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/S0020-0255(02)00411-5","volume":"152","author":"KR Bhutani","year":"2003","unstructured":"Bhutani KR, Rosenfeld A (2003) Strong arcs in fuzzy graphs. Inf Sci 152:319\u2013322","journal-title":"Inf Sci"},{"key":"7636_CR10","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/j.fss.2020.05.014","volume":"410","author":"JY Ma","year":"2021","unstructured":"Ma JY, Li QG, Zhou XN (2021) Fuzzy edge connectivity and fuzzy local edge connectivity with applications to communication networks. Fuzzy Sets Syst 410:109\u2013125","journal-title":"Fuzzy Sets Syst"},{"issue":"1","key":"7636_CR11","first-page":"145","volume":"16","author":"OT Manjusha","year":"2019","unstructured":"Manjusha OT, Sunitha MS (2019) Coverings, matchings and paired domination in fuzzy graphs using strong arcs. Iranian J Fuzzy Syst 16(1):145\u2013157","journal-title":"Iranian J Fuzzy Syst"},{"issue":"11","key":"7636_CR12","doi-asserted-by":"publisher","first-page":"1760","DOI":"10.1016\/j.ins.2009.01.003","volume":"179","author":"S Mathew","year":"2009","unstructured":"Mathew S, Sunitha MS (2009) Types of arcs in a fuzzy graph. Inf Sci 179(11):1760\u20131768","journal-title":"Inf Sci"},{"issue":"4","key":"7636_CR13","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1016\/j.ins.2009.10.006","volume":"180","author":"S Mathew","year":"2010","unstructured":"Mathew S, Sunitha MS (2010) Node connectivity and arc connectivity of a fuzzy graph. Inf Sci 180(4):519\u2013531","journal-title":"Inf Sci"},{"issue":"6","key":"7636_CR14","doi-asserted-by":"publisher","first-page":"1096","DOI":"10.1109\/TFUZZ.2013.2243154","volume":"21","author":"S Mathew","year":"2013","unstructured":"Mathew S, Sunitha MS (2013) Strongest strong cycles and $$\\theta -$$fuzzy graphs. IEEE Trans Fuzzy Syst 21(6):1096\u20131104","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"7636_CR15","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1016\/j.fss.2017.10.004","volume":"352","author":"S Mathew","year":"2018","unstructured":"Mathew S, Anjali N, Mordeson JN (2018) Transitive blocks and their applications in fuzzy interconnection networks. Fuzzy Sets Syst 352:142\u2013160","journal-title":"Fuzzy Sets Syst"},{"key":"7636_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-71407-3","volume-title":"Fuzzy Graph Theory","author":"S Mathew","year":"2018","unstructured":"Mathew S, Mordeson JN, Malik DS (2018) Fuzzy Graph Theory. Springer"},{"key":"7636_CR17","doi-asserted-by":"crossref","unstructured":"Mathew S, Yang HL, Mathew JK (2018) Saturation in fuzzy graphs. New Math Nat Comput 14:113\u2013128","DOI":"10.1142\/S1793005718500084"},{"issue":"1","key":"7636_CR18","first-page":"39","volume":"2","author":"A Nagoorgani","year":"2009","unstructured":"Nagoorgani A, Malarvizhi J (2009) Isomorphism properties of strong fuzzy graphs. Int J Algorithms, Comput Math 2(1):39\u201347","journal-title":"Int J Algorithms, Comput Math"},{"key":"7636_CR19","first-page":"33","volume":"12","author":"A Nagoorgani","year":"2008","unstructured":"Nagoorgani A, Radha K (2008) On regular fuzzy graphs. J Phys Sci 12:33\u201340","journal-title":"J Phys Sci"},{"issue":"4","key":"7636_CR20","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1007\/s13042-014-0267-8","volume":"7","author":"A Nagoorgani","year":"2016","unstructured":"Nagoorgani A, Akram M, Vijayalakshmi P (2016) Certain types of fuzzy sets in a fuzzy graph. Int J Mach Learn Cybern 7(4):573\u2013579","journal-title":"Int J Mach Learn Cybern"},{"key":"7636_CR21","doi-asserted-by":"publisher","first-page":"803","DOI":"10.1016\/0167-8655(94)90008-6","volume":"15","author":"PS Nair","year":"1994","unstructured":"Nair PS (1994) Triangle and parallelogram laws on fuzzy graphs. Pattern Recognit Lett 15:803\u2013805","journal-title":"Pattern Recognit Lett"},{"key":"7636_CR22","doi-asserted-by":"publisher","first-page":"5119","DOI":"10.1007\/s00500-019-04265-z","volume":"24","author":"S Poulik","year":"2020","unstructured":"Poulik S, Ghorial G (2020) Certain indices of graphs under bipolar fuzzy environment with applications. Soft Comput 24:5119\u20135131","journal-title":"Soft Comput"},{"key":"7636_CR23","doi-asserted-by":"publisher","first-page":"608","DOI":"10.1016\/j.ins.2020.09.050","volume":"545","author":"S Poulik","year":"2021","unstructured":"Poulik S, Ghorial G (2021) Determination of journeys order based on graph\u2019s wiener absolute index with bipolar fuzzy information. Inf Sci 545:608\u2013619","journal-title":"Inf Sci"},{"key":"7636_CR24","doi-asserted-by":"publisher","first-page":"1115","DOI":"10.1007\/s40747-021-00580-x","volume":"8","author":"S Poulik","year":"2022","unstructured":"Poulik S, Ghorial G (2022) Applications of graph\u2019s complete degree with bipolar fuzzy information. Complex Intell Syst 8:1115\u20131127","journal-title":"Complex Intell Syst"},{"key":"7636_CR25","doi-asserted-by":"publisher","first-page":"4557","DOI":"10.1007\/s10462-021-10111-2","volume":"55","author":"S Poulik","year":"2022","unstructured":"Poulik S, Ghorial G (2022) Estimation of most effected cycles and busiest network route based on complexity function of graph in fuzzy environmen. Artif Intell Rev 55:4557\u20134574","journal-title":"Artif Intell Rev"},{"key":"7636_CR26","doi-asserted-by":"publisher","first-page":"549","DOI":"10.1007\/s41066-021-00284-0","volume":"7","author":"S Poulik","year":"2022","unstructured":"Poulik S, Ghorial G (2022) New concepts of inverse fuzzy mixed graphs and its application. Granular Comput 7:549\u2013559","journal-title":"Granular Comput"},{"key":"7636_CR27","doi-asserted-by":"publisher","first-page":"711","DOI":"10.1007\/s00500-020-05180-4","volume":"25","author":"S Poulik","year":"2021","unstructured":"Poulik S, Ghorial G, Xin Q (2021) Pragmatic results in taiwan education system based ivfg and ivng. Soft Comput 25:711\u2013724","journal-title":"Soft Comput"},{"key":"7636_CR28","volume-title":"Fuzzy graphs. fuzzy sets and their applications to cognitive and decision processes","author":"A Rosenfeld","year":"1975","unstructured":"Rosenfeld A (1975) Fuzzy graphs. fuzzy sets and their applications to cognitive and decision processes. Academic Press, New York"},{"key":"7636_CR29","doi-asserted-by":"publisher","first-page":"3417","DOI":"10.1007\/s00500-021-05610-x","volume":"25","author":"M Sarwar","year":"2021","unstructured":"Sarwar M, Akram M, Shahzadi S (2021) Bipolar fuzzy soft information applied to hypergraphs. Soft Comput 25:3417\u20133439","journal-title":"Soft Comput"},{"issue":"3\u20134","key":"7636_CR30","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1016\/S0020-0255(98)10066-X","volume":"113","author":"MS Sunitha","year":"1999","unstructured":"Sunitha MS, Vijayakumar A (1999) A characterization of fuzzy trees. Inf Sci 113(3\u20134):293\u2013300","journal-title":"Inf Sci"},{"issue":"1","key":"7636_CR31","first-page":"13","volume":"13","author":"MS Sunitha","year":"2005","unstructured":"Sunitha MS, Vijayakumar A (2005) Blocks in fuzzy graphs. J Fuzzy Math 13(1):13\u201323","journal-title":"J Fuzzy Math"},{"key":"7636_CR32","volume-title":"Fuzzy relations, fuzzy graphs and their applications to clustering analysis .fuzzy sets and their applications","author":"RT Yeh","year":"1975","unstructured":"Yeh RT, Bang SY (1975) Fuzzy relations, fuzzy graphs and their applications to clustering analysis .fuzzy sets and their applications. Academic Press, New York"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-022-07636-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-022-07636-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-022-07636-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T07:10:49Z","timestamp":1674803449000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-022-07636-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,16]]},"references-count":32,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["7636"],"URL":"https:\/\/doi.org\/10.1007\/s00500-022-07636-1","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,16]]},"assertion":[{"value":"17 October 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 November 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"There is no conflict of interest between the authors and the institute where the work has been carried out.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"The article does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}]}}