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

For any real $α\in [0,1]$, Nikiforov defined the $A_α$-matrix of a graph $G$ as $A_α(G)=αD(G)+(1-α)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. The largest eigenvalue of $A_α(G)$ is called the $α$-index or the $A_α$-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 $α$-index for $α\in [\frac{1}{2},1)$ among all minimally 2-connected graphs with given order or size, respectively.

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