Sharp bounds on the $A_{\alpha}$-index of graphs in terms of the independence number

TL;DR

This paper establishes bounds on the $A_{\alpha}$-index of graphs based on their independence number, characterizing extremal graphs for various parameters and extending previous spectral graph theory results.

Contribution

It provides the first comprehensive characterization of graphs with extremal $A_{\alpha}$-index for specific independence numbers across different $\alpha$ ranges.

Findings

Identifies graphs with minimum $A_{\alpha}$-index for various independence numbers.

Determines unique graphs with maximum $A_{\alpha}$-index given the independence number.

Extends spectral bounds to a broader class of graphs and parameters.

Abstract

Given a graph $G$, the adjacency matrix and degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} proposed the $A_{\alpha}$-matrix: $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G),$ where $\alpha\in [0, 1]$. The largest eigenvalue of this novel matrix is called the $A_\alpha$-index of $G$. In this paper, we characterize the graphs with minimum $A_\alpha$-index among $n$-vertex graphs with independence number $i$ for $\alpha\in[0,1)$, where $i=1,\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil,{\lfloor\frac{n}{2}\rfloor+1},n-3,n-2,n-1,$ whereas for $i=2$ we consider the same problem for $\alpha\in [0,\frac{3}{4}{]}.$ Furthermore, we determine the unique graph (resp. tree) on $n$ vertices with given independence number having the maximum $A_\alpha$-index with $\alpha\in[0,1)$, whereas for the $n$-vertex bipartite graphs with given independence number, we characterize the unique graph having the maximum $A_\alpha$-index with $\alpha\in[\frac{1}{2},1).$