Recovering quantum graphs from their Bloch spectrum

TL;DR

This paper introduces the Bloch spectrum of quantum graphs and demonstrates its ability to uniquely determine key structural properties, including planarity and 3-connectedness, thus enabling graph reconstruction.

Contribution

It is the first to show that the Bloch spectrum fully determines planar 3-connected quantum graphs and related geometric features.

Findings

Bloch spectrum determines the Albanese torus.

It reveals the block structure and planarity of the graph.

The spectrum enables reconstruction of planar 3-connected quantum graphs.

Abstract

We define the Bloch spectrum of a quantum graph to be the collection of the spectra of a family of Schrodinger operators parametrized by the cohomology of the quantum graph. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum completely determines planar 3-connected quantum graphs.