On spectral radii of unraveled balls

TL;DR

This paper establishes lower bounds on the spectral radius of unraveled balls in graphs, linking local structure to eigenvalues, with implications for graph spectral theory.

Contribution

It provides new lower bounds on the spectral radius of unraveled balls and relates local graph modifications to eigenvalue estimates.

Findings

Lower bound on maximum spectral radius of unraveled balls

Second largest eigenvalue bound based on local average degree

Connection between local graph structure and spectral properties

Abstract

Given 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 prove a lower bound on the maximum spectral radius of unraveled balls of fixed radius, and we show, among other things, that if the average degree of $G$ after deleting any ball of radius $r$ is at least $d$ then its second largest eigenvalue is at least $2\sqrt{d-1}\cos(\frac{\pi}{r+1})$.