A Frucht's theorem for quantum graphs

TL;DR

This paper extends Frucht's theorem to quantum graphs, demonstrating that every group can be realized as the symmetry group of a quantum graph by defining an appropriate notion of symmetry for these structures.

Contribution

It introduces a quantum graph analogue of Frucht's theorem, establishing a correspondence between groups and quantum graph symmetries.

Findings

Every group can be realized as a quantum graph symmetry group

Defines a notion of symmetry for quantum graphs

Establishes an analogue of Frucht's theorem for quantum graphs

Abstract

A celebrated theorem due to R. Frucht states that, roughly speaking, each group is isomorphic to the symmetry group of some graph. By "symmetry group" the group of all graph automorphisms is meant. We provide an analogue of this result for quantum graphs, i.e., for Schr\"odinger equations on a metric graph, after suitably defining the notion of symmetry.