Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap

TL;DR

This paper establishes new upper bounds on the spectral gap of Laplacians in quantum graphs based on geometric properties, revealing limitations of certain metrics in predicting spectral behavior.

Contribution

It introduces novel metric quantities like avoidance diameter and analyzes their effectiveness in bounding the spectral gap, connecting graph geometry with spectral properties.

Findings

New upper bounds on spectral gap using geometric metrics

Introduction of avoidance diameter as a novel metric

Limitations of certain metrics in predicting spectral bounds

Abstract

We derive several upper bounds on the spectral gap of the Laplacian with standard or Dirichlet vertex conditions on compact metric graphs. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoidance diameter. Using known results about Ramanujan graphs, a class of expander graphs, we also prove that some of these metric quantities, or combinations thereof, do not to deliver any spectral bounds with the correct scaling.