Bounds on Geometric Eigenvalues of Graphs

TL;DR

This paper investigates bounds on a generalized eigenvalue of graphs, $mbda(G, X)$, which extends the classical eigenvalue concept by embedding graph vertices into metric spaces, revealing non-monotonic behaviors under graph modifications.

Contribution

It provides new bounds for $mbda(G, X)$ and $mbda(G, H)$, generalizing existing results, and explores how these eigenvalues are affected by changes in the graph or metric space.

Findings

$mbda(G, H)$ is not monotone in $G$ or $H$

Established bounds for $mbda(G, X)$ and $mbda(G, H)$

Generalized classical eigenvalue results to metric space embeddings

Abstract

The smallest nonzero eigenvalue of the normalized Laplacian matrix of a graph has been extensively studied and shown to have many connections to properties of the graph. We here study a generalization of this eigenvalue, denoted $\lambda(G, X)$, introduced by Mendel and Naor in 2010, obtained by embedding the vertices of the graph $G$ into a metric space $X$. We consider general bounds on $\lambda(G, X)$ and $\lambda(G, H)$, where $H$ is a graph under the standard distance metric, generalizing some existing results for the standard eigenvalue. We consider how $\lambda(G, H)$ is affected by changes to $G$ or $H$, and show $\lambda(G, H)$ is not monotone in either $G$ or $H$.