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The applications of directed HBVFGs with algorithms for finding the optimal path in a network and the dominant node and influence of index with the self-persistence degree of a node in a social network are presented. For each problem, an algorithm is developed and its effectiveness is demonstrated by examples. The proposed concept is assessed in terms of theory and practice. The benefits of the proposed solution are highlighted and a clear comparison is made with the existing methods.<\/jats:p>","DOI":"10.1007\/s40747-024-01438-8","type":"journal-article","created":{"date-parts":[[2024,4,20]],"date-time":"2024-04-20T06:02:01Z","timestamp":1713592921000},"page":"5195-5211","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Fuzzy graphs and their applications in finding the best route, dominant node and influence index in a network under the hesitant bipolar-valued fuzzy environment"],"prefix":"10.1007","volume":"10","author":[{"given":"Jambi Ratna","family":"Raja","sequence":"first","affiliation":[]},{"given":"Jeong Gon","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Dhanraj","family":"Dhotre","sequence":"additional","affiliation":[]},{"given":"Pravin","family":"Mane","sequence":"additional","affiliation":[]},{"given":"Omprakash Sugdeo","family":"Rajankar","sequence":"additional","affiliation":[]},{"given":"Antonios","family":"Kalampakas","sequence":"additional","affiliation":[]},{"given":"Navin D.","family":"Jambhekar","sequence":"additional","affiliation":[]},{"given":"D. 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