On the $\alpha$-index of minimally 2-connected graphs with given order or size

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.