On the Spectral Gap of a Quantum Graph

TL;DR

This paper derives bounds on the spectral gap of the Laplacian on metric graphs based on geometric and combinatorial properties, revealing limitations and providing sharp estimates that sometimes outperform purely combinatorial bounds.

Contribution

It establishes universal isoperimetric bounds for the spectral gap of quantum graphs, linking spectral properties to geometric and combinatorial parameters, and compares these with bounds for the normalized Laplacian.

Findings

No diameter-only bound exists for the spectral gap.

Sharp estimates are obtained in terms of length, diameter, vertices, and edges.

Some bounds for the normalized Laplacian are sharper than existing combinatorial results.

Abstract

We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.