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

Abstract

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

$$\begin{aligned} A_\alpha (G) = \alpha D(G) + (1-\alpha ) A(G) \quad \text {for any }\alpha \in [0,1]. \end{aligned}$$

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.