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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) = \alphaD(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 $A_{\alpha}$-spectralradius of $G$. In this article, we characterize the graphs with maximum$A_{\alpha}$-spectral radius among the class of unicyclic and bicyclic graphsof order $n$ with fixed girth $g$. Also, we identify the unique graphs withmaximum $A_{\alpha}$-spectral radius among the class of unicyclic and bicyclicgraphs of order $n$ with $k$ pendant vertices.

On the maximum $A_{\alpha}$-spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices