A short note on $A_\alpha$-eigenvalues for simple graphs

TL;DR

This paper compares two lower bounds for the spectral radius of the $A_eta$ matrix in simple graphs, showing that one bound is superior when the graph has no isolated nodes.

Contribution

It provides a comparison and proof of the superiority of one lower bound over another for the $A_eta$-spectral radius in graphs without isolated nodes.

Findings

One lower bound is better than the other when no isolated nodes are present.

The comparison clarifies the conditions under which each bound is optimal.

Results improve understanding of spectral properties of $A_eta$ matrices.

Abstract

Given a simple graph $G$, its $A_\alpha$ matrix is a convex combination with parameter $\alpha\in [0,1]$ of its adjacency matrix and its degree diagonal matrices. Here we compare two lower bounds presented in [J. D. G. Silva Jr., C. S. Oliveira and L. M. G. C. Costa. "Some results involving the $A_\alpha$-eigenvalues for graphs and line graphs"] for the spectral radius of $A_\alpha$, and prove that one is better than the other when there are no isolated nodes in $G$.