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