On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

TL;DR

This paper establishes bounds on the spectral radius of unraveled balls and the normalized Laplacian eigenvalues in weighted graphs, providing insights into spectral properties of irregular graphs and extending classical bounds.

Contribution

It introduces a lower bound on the spectral radius of unraveled balls and an upper bound on normalized Laplacian eigenvalues for irregular graphs, generalizing Alon--Boppana bounds.

Findings

Lower bound on spectral radius of unraveled balls in weighted graphs

Upper bound on the s-th smallest normalized Laplacian eigenvalue for irregular graphs

Extension of Alon--Boppana type bounds to irregular graph classes

Abstract

For a graph $G$, the unraveled ball of radius $r$ centered at a vertex $v$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s$-th (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon--Boppana type bound for a class of irregular graphs.