  For a connected graph $G$, let $A(G)$ be the adjacency matrix of $G$ and
$D(G)$ be the diagonal matrix of the degrees of the vertices in $G$. The
$A_{\alpha}$-matrix of $G$ is defined as \begin{align*} A_\alpha (G) = \alpha
D(G) + (1-\alpha) A(G) \quad \text{for any $\alpha \in [0,1]$}. \end{align*}
  The largest eigenvalue of $A_{\alpha}(G)$ is called the $A_{\alpha}$-spectral
radius of $G$. In this article, we characterize the graphs with maximum
$A_{\alpha}$-spectral radius among the class of unicyclic and bicyclic graphs
of order $n$ with fixed girth $g$. Also, we identify the unique graphs with
maximum $A_{\alpha}$-spectral radius among the class of unicyclic and bicyclic
graphs of order $n$ with $k$ pendant vertices.
