20.500.12556/DiRROS-23751 Energy and Laplacian Energy of Graphs under the Bipolar Valued Hesitant Fuzzy Framework Bipolar-valued hesitant fuzzy graphs (BVHFGs) provide a suitable framework for representing knowledge in situations characterized by uncertainty, imprecision, and hesitations. Current research shows a lack of studies on energy in contexts involving bipolarity, hesitations, and fuzzy data, motivating us to propose new definitions of energy in this area. In this study, we introduce innovative notions of graph energy and Laplacian energy within the framework of a bipolar valued hesitant fuzzy setting and scrutinize certain characteristics and various types of bounds of these concepts. Additionally, the investigation explores the interplay between the energy and Laplacian energy of BVHFGs. Consequently, a numerical illustration is provided, encompassing the identification of optimal alternatives to elucidate the pragmatic application of the proposed theoretical frameworks within the realm of decision-making. This empirical demonstration underscores the efficacy and relevance of the developed methodology in addressing real-world decision-making challenges. fuzzy graph bipolar valued hesitant fuzzy graph Spectrum energy Laplacian energy mehki graf bipolarno vrednoten oklevajoč mehki graf spekter energija Laplaceova energija true false true Islamic Azad University Angleški jezik Slovenski jezik © By the Authors. Neznano 2025-10-01 03:56:30 2026-09-01 13:21:24 2026-09-02 04:15:54 0000-00-00 00:00:00 2026 0 0 Nasl. z nasl. zaslona; Opis vira z dne 30. 9. 2025; Soavtorji: Abhay S. Ranadive, Sovan Samanta, Leo Mrsic, Antonios Kalampakas; str. 26-53 no. 1 Vol. 5 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-05-07 51 2821-0131 251180547 https://sanad.iau.ir/Journal/tfss/Article/1196801 1 456db46f-a5f7-11f1-8a94-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=50290 RAZ_Pandey_Sakshi_Dev_2026.pdf RAZ_Pandey_Sakshi_Dev_2026.pdf 1 3EBF929CE604DC2A276FBB95299822AD db4cb6244fabe9e348bc47419c81157cd6279e11b94b32ded5e2098a3c434d1a 69fbfcd6-a5f7-11f1-8a94-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=50293 Rudolfovo – Znanstveno in tehnološko središče Novo mesto 0 0 0