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

Giovanni Barbarino

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

Combinatorics 2026-01-27 v1

Authors: Giovanni Barbarino

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 $G$ .

Keywords

graph laplacian spectral graph theory graph theory

Cite

@article{arxiv.2601.18365,
  title  = {A short note on $A_\alpha$-eigenvalues for simple graphs},
  author = {Giovanni Barbarino},
  journal= {arXiv preprint arXiv:2601.18365},
  year   = {2026}
}