In an earlier work, finite groups whose power graphs are minimally edge connected have been classified. In this article, first we obtain a necessary and sufficient condition for an arbitrary graph to be minimally edge connected. Consequently, we characterize finite groups whose enhanced power graphs and order superpower graphs, respectively, are minimally edge connected. Moreover, for a finite non-cyclic group $G$, we prove that $G$ is an elementary abelian $2$-group if and only if its enhanced power graph is minimally connected. Also, we show that $G$ is a finite $p$-group if and only if its order superpower graph is minimally connected. Finally, we characterize all the finite nilpotent groups such that the minimum degree and the vertex connectivity of their order superpower graphs are equal.