  Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the
diagonal matrix of vertex degrees of $G$. For any real $\alpha \in [0,1]$,
Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha
D(G)+(1-\alpha)A(G)$. The largest eigenvalue of $A_\alpha(G)$ is called the
$\alpha$-index or the $A_\alpha$-spectral radius of $G$. A graph is minimally
$k$-(edge)-connected if it is $k$-(edge)-connected and deleting any arbitrary
chosen edge always leaves a graph which is not $k$-(edge)-connected. In this
paper, we characterize the minimally 2-edge-connected graphs and minimally
3-connected graph with given order having the maximum $\alpha$-index for
$\alpha \in [\frac{1}{2},1)$, respectively.
