Proof of an open problem on the Sombor index.

The Sombor index is one of the geometry-based descriptors, which was defined as S O (G) = ∑ u v ∈ E (G) d u 2 + d v 2 , where d u (resp. d v ) denotes the degree of vertex u (resp. v) in G. In this note, we determine the graphs among the set of graphs with vertex connectivity (resp. edge connectivity) at most k having the maximum and minimum Sombor indices, which solves an open problem on the Sombor index proposed by Hayat and Rehman [On Sombor index of graphs with a given number of cut-vertices, MATCH Commun. Math. Comput. Chem. 89 (2023) 437–450]. For some conclusions of the above paper, we first give some counterexamples, then provide another simple proof about the minimum Sombor indices of graphs with n vertices, k cut vertices and at least one cycle. [ABSTRACT FROM AUTHOR]