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

TL;DR

This paper characterizes graphs with the minimum and maximum $A_\alpha$ spectral radius among those with independence number $n-4$, for $\alpha$ in the range [1/2, 1), extending previous spectral graph theory results.

Contribution

It provides a complete characterization of extremal graphs for the $A_\alpha$ spectral radius in the class with independence number $n-4$ for a specific range of $\alpha$, which was not previously known.

Findings

Identifies graphs with minimum $A_\alpha$ spectral radius in $\mathcal{G}_{n,n-4}$ for $\alpha \in [1/2,1)$.

Identifies graphs with maximum $A_\alpha$ spectral radius in $\mathcal{G}_{n,n-4}$ for $\alpha \in [1/2,1)$.

Extends previous results on spectral extremal problems to the $A_\alpha$ matrix for a specific independence number.

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.