The $A_{\alpha}$-spectral Radius of Bicyclic Graphs with Given Degree Sequences

Abstract

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$, respectively. For any real $\alpha \in [0,1]$, Nikiforov defined the matrix $A_{\alpha}(G)$ as \[ A_{\alpha}(G) = \alpha D(G) + (1-\alpha) A(G). \] In this paper, we generalize some previous results about the $A_{1/2}$-spectral radius of bicyclic graphs with a given degree sequence. Furthermore, we characterize all extremal bicyclic graphs which have the largest $A_{\alpha}$-spectral radius in the set of all bicyclic graphs with prescribed degree sequences.