Spectral gap for quantum graphs and their connectivity

TL;DR

This paper explores how the spectral gap of Laplace operators on metric graphs is influenced by their connectivity and geometric properties, revealing complex relationships and providing explicit criteria for how edge modifications affect the spectral gap.

Contribution

It demonstrates that the spectral gap's dependence on graph connectivity is not straightforward and introduces explicit criteria for how edge size changes impact the spectral gap.

Findings

Adding large edges decreases the spectral gap.

Removing small edges decreases the spectral gap.

The spectral gap depends on both topology and geometry.

Abstract

The spectral gap for Laplace operators on metric graphs is investigated in relation to graph's connectivity, in particular what happens if an edge is added to (or deleted from) a graph. It is shown that in contrast to discrete graphs connection between the connectivity and the spectral gap is not one-to-one. The size of the spectral gap depends not only on the topology of the metric graph but on its geometric properties as well. It is shown that adding sufficiently large edges as well as cutting away sufficiently small edges leads to a decrease of the spectral gap. Corresponding explicit criteria are given.