  Recently in 2021, Gutman introduced the Sombor index of a graph, a novel
degree-based topological index. It has been shown that the Sombor index
efficiently models the thermodynamic properties of chemical compounds. Assume
$\mathbb{B}_n^k$ (resp. $\mathbb{V}_n^k$) comprises all graphs with order $n$
having number of bridges (resp. vertex-connectivity) $k$. Horoldagva & Xu
(2021) characterized graphs achieving the maximum Sombor index of graphs in
$\mathbb{B}_n^k$. This paper characterizes graphs achieving the minimum Sombor
index in $\mathbb{B}_n^k$. Certain auxiliary operation on graphs in
$\mathbb{B}_n^k$ are introduced and employed for the characterization.
Moreover, we characterize graphs achieving maximum Sombor index in
$\mathbb{V}_n^k$. ome open problems, which naturally arise from this work, have
been proposed at the end.
