  Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D
(G)$. In 2017, Nikiforov [Appl. Anal. Discrete Math., 11 (2017) 81--107]
defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for any real
$\alpha\in[0,1]$. The largest eigenvalue of $A(G)$ is called the spectral
radius of $G$, while the largest eigenvalue of $A_\alpha(G)$ is called the
$A_\alpha$ spectral radius of $G$. Let $\mathcal{G}_{n,i}$ be the set of graphs
of order $n$ with independence number $i$. Recently, for all graphs in
$\mathcal{G}_{n,i}$ having the minimum or the maximum $A$, $Q$ and $A_\alpha$
spectral radius where
$i\in\{1,2,\lfloor\frac{n}{2}\rfloor\,\lceil\frac{n}{2}\rceil+1,n-3,n-2,n-1\}$,
there are some results have been given by Xu, Li and Sun et al., respectively.
In 2021, Luo and Guo [Discrete Math., 345 (2022) 112778] determined all graphs
in $\mathcal{G}_{n,n-4}$ having the minimum spectral radius. In this paper, we
characterize the graphs in $\mathcal{G}_{n,n-4}$ having the minimum and the
maximum $A_\alpha$ spectral radius for $\alpha\in[\frac{1}{2},1)$,
respectively.
