  Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D
(G)$. In 2017, Nikiforov defined the matrix $A_\alpha(G) = \alpha D(G) +
(1-\alpha)A(G)$ for any real $\alpha\in[0,1]$. The largest eigenvalue of
$A_\alpha(G)$ is called the $A_\alpha$ spectral radius or the $A_\alpha$-index
of $G$.
  Let $\mathcal{G}_{n,k}^d$ be the set of $k$-connected graphs of order $n$
with diameter $d$. In this paper, we determine the graphs with maximum
$A_\alpha$ spectral radius among all graphs in $\mathcal{G}_{n,k}^d$ for any
$\alpha\in[0,1)$, where $k\geq2$ and $d\geq2$. We generalizes the results about
adjacency matrix of Theorem 3.6 in [P. Huang, W.C. Shiu, P.K. Sun, Linear
Algebra Appl., 488 (2016) 350--362] and the results about signless Laplacian
matrix of Theorem 3.4 in [P. Huang, J.X. Li, W.C. Shiu, Linear Algebra Appl.,
617 (2021) 78--99]. Furthermore, we also obtain the upper and lower bounds of
the extremal graph in $\mathcal{G}_{n,k}^d$.
