The $A_{\alpha }$ spectral radius with given independence number $n-4$

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$. In 2017, Nikiforov [Appl. Anal. Discrete Math., 11 (2017) 81--107] defined the matrix $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$ for any real $\alpha \in [0,1]$. The largest eigenvalue of $A(G)$ is called the spectral radius of $G$, while the largest eigenvalue of $A_{\alpha }(G)$ is called the $A_{\alpha }$ spectral radius of $G$. Let ${\mathcal{G}}_{n,i}$ be the set of graphs of order $n$ with independence number $i$. Recently, for all graphs in ${\mathcal{G}}_{n,i}$ having the minimum or the maximum $A$, $Q$ and $A_{\alpha }$ spectral radius where $i\in \{1,2,\lfloor \frac{n}{2}\rfloor \lceil \frac{n}{2}\rceil +1,n-3,n-2,n-1\}$, there are some results have been given by Xu, Li and Sun et al., respectively. In 2021, Luo and Guo [Discrete Math., 345 (2022) 112778] determined all graphs in ${\mathcal{G}}_{n,n-4}$ having the minimum spectral radius. In this paper, we characterize the graphs in ${\mathcal{G}}_{n,n-4}$ having the minimum and the maximum $A_{\alpha }$ spectral radius for $\alpha \in [\frac{1}{2},1)$, respectively.