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

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 \(A_{\alpha }\) $A_{\alpha }$ -matrix of G is defined as \(\begin{aligned} A_\alpha (G) = \alpha D(G) + (1-\alpha ) A(G) \quad \text {for any }\alpha \in [0,1]. \end{aligned}\) $$\begin{array}{c}\hfill A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)\text{for any}\alpha \in [0,1].\end{array}$$ The largest eigenvalue of \(A_{\alpha }(G)\) $A_{\alpha }(G)$ is called the \(A_{\alpha }\) $A_{\alpha }$ -spectral radius of G. In this article, we characterize the graphs with maximum \(A_{\alpha }\) $A_{\alpha }$ -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 \(A_{\alpha }\) $A_{\alpha }$ -spectral radius among the class of unicyclic and bicyclic graphs of order n with k pendant vertices.