A short note on $A_{\hat{I}}\pm$-eigenvalues for simple graphs

Abstract

Given a simple graph $G$, its $A_{\hat{I}}\pm$ matrix is a convex combination with parameter $\hat{I}\pm \in [0,1]$ 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 $A_{\hat{I}}\pm$-eigenvalues for graphs and line graphs"] for the spectral radius of $A_{\hat{I}}\pm$, and prove that one is better than the other when there are no isolated nodes in $G$.