On the $\alpha$-index of minimally $k$-(edge-)connected graphs for small $k$

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.