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The Sombor index, which Gutman proposed recently in 2021, is a graph-based degree-related topological descriptor with potential applicability in understanding how compounds behave thermodynamically. Let Vνρ (resp. Eνρ) denote the collection of all ν -vertex connected graphs having number of cut-vertices (resp. edge-connectivity) ρ . In Problem 1 of [On Sombor index of graphs with a given number of cut-vertices, MATCH Commun. Math. Comput. Chem., 89 (2023) 437–450], the authors asked to find maximum Sombor index of graph in Vνρ . Moreover, in Remark 1 of [On the Sombor index of graphs with given connectivity and number of bridges, arXiv:2208.09993 , (2022)], the authors raise a conjecture on the maximum Sombor index in Eνρ . This paper solves both of the open problems and find sharp upper bounds on the Sombor index of graphs in Vνρ and Eνρ . The respective maximum graphs achieving the bounds have also been classified. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.