A transfer principle and applications to eigenvalue estimates for graphs

TL;DR

This paper extends the transfer principle to bound eigenvalues of graphs based on genus, providing tight bounds and applications to mesh partitioning, improving previous results in spectral graph theory.

Contribution

It introduces a variant of the transfer principle that relates eigenvalues of graphs to their genus, with tight bounds and broad applications.

Findings

Eigenvalue bounds depend on genus and maximum degree.

The bounds are tight up to a constant factor.

Application to mesh partitioning extends previous work.

Abstract

In this paper, we prove a variant of the Burger-Brooks transfer principle which, combined with recent eigenvalue bounds for surfaces, allows to obtain upper bounds on the eigenvalues of graphs as a function of their genus. More precisely, we show the existence of a universal constants $C$ such that the $k$-th eigenvalue $\lambda_k^{nr}$ of the normalized Laplacian of a graph $G$ of (geometric) genus $g$ on $n$ vertices satisfies $$\lambda_k^{nr}(G) \leq C \frac{d_{\max}(g+k)}{n},$$ where $d_{\max}$ denotes the maximum valence of vertices of the graph. This result is tight up to a change in the value of the constant $C$, and improves recent results of Kelner, Lee, Price and Teng on bounded genus graphs. To show that the transfer theorem might be of independent interest, we relate eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple graph models, and discuss an application to the mesh partitioning problem, extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang to arbitrary meshes.