The $A_{\alpha }$-spectral radius of graphs with given degree sequence

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$, and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\alpha \in [0,1]$, write $A_{\alpha }(G)$ for the matrix $$A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G).$$ This paper presents some extremal results about the spectral radius $\rho (A_{\alpha }(G))$ of $A_{\alpha }(G)$ that generalize previous results about $\rho (A_0(G))$ and $\rho (A_{\frac{1}{2}}(G))$. In this paper, we give some results on graph perturbation for $A_{\alpha }$-matrix with $\alpha \in [0,1)$. As applications, we characterize all extremal trees with the maximum $A_{\alpha }$-spectral radius in the set of all trees with prescribed degree sequence firstly. Furthermore, we characterize the unicyclic graphs that have the largest $A_{\alpha }$-spectral radius for a given unicycilc degree sequence.