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

TL;DR

This paper investigates the maximum spectral radius of a family of matrices $A_eta(G)$ interpolating between degree and adjacency matrices, providing extremal characterizations for trees and unicyclic graphs with given degree sequences.

Contribution

It generalizes previous spectral radius results for specific $eta$ values and characterizes extremal graphs with maximum $A_eta$-spectral radius for given degree sequences.

Findings

Characterization of extremal trees with maximum $A_eta$-spectral radius.

Identification of unicyclic graphs with largest $A_eta$-spectral radius.

Extension of spectral radius bounds for $A_eta$ matrices with $eta eq 0, 0.5$.

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.