In this paper, we show that in the class of graphs of order n and given (vertex or edge) connectivity equal to $k$ (or at most equal to $k$), $1\leq k\leq n-1$, the graph $K_k+(K_1\cup K_{n-k-1})$ is the unique graph such that zeroth-order general Randić index, general sum-connectivity index and general Randić connectivity index are maximum and general hyper-Wiener index is minimum provided $\alpha\geq1$. Also, for 2-connected (or 2-edge connected) graphs and $\alpha>0$ the unique graph minimizing these indices is the $n$-vertex cycle $C_n$.