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

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 $G$.