  Given a graph $G,$ a subset of vertices is called a maximum dissociation set
of $G$ if it induces a subgraph with vertex degree at most 1, and the subset
has maximum cardinality. The cardinality of a maximum dissociation set is
called the dissociation number of $G$. The adjacency matrix and the degree
diagonal matrix of $G$ are denoted by $A(G)$ and $D(G),$ respectively. In 2017,
Nikiforov proposed the $A_\alpha$-matrix: $A_\alpha(G)=\alpha
D(G)+(1-\alpha)A(G),$ where $\alpha\in[0,1].$ The largest eigenvalue of this
novel matrix is called the $A_\alpha$-index of $G.$ In this paper, we firstly
determine the connected graph (resp. bipartite graph, tree) having the largest
$A_\alpha$-index over all connected graphs (resp. bipartite graphs, trees) with
fixed order and dissociation number. Secondly, we describe the structure of all
the $n$-vertex graphs having the minimum $A_\alpha$-index with dissociation
number $\tau$, where $\tau\geqslant\lceil\frac{2}{3}n\rceil.$ Finally, we
identify all the connected $n$-vertex graphs with dissociation number
$\tau\in\{2,\lceil\frac{2}{3}n\rceil,n-1,n-2\}$ having the minimum
$A_\alpha$-index.
