A 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.