Voltage quantum graphs and a Gross-Tucker theorem for quantum graphs

TL;DR

This paper extends the classical voltage graph construction to quantum graphs and finite abelian groups, providing a quantum version of the Gross-Tucker theorem that characterizes derived graphs.

Contribution

It generalizes voltage graph constructions to quantum graphs and proves a quantum Gross-Tucker theorem for characterizing derived graphs.

Findings

Quantum graphs can be constructed from classical voltage graphs.

Quantum isomorphism between quantum and classical graphs can be achieved.

A quantum version of the Gross-Tucker theorem is established.

Abstract

A voltage graph is a finite directed graph whose edges are labeled by elements of a finite group $G$. A classical construction of Gross and Tucker associates to every voltage graph with vertex set $V$ a so-called derived graph with vertex set $V \times G$. We generalize their construction to quantum graphs and finite abelian groups. Remarkably, the construction can produce true quantum graphs starting from a classical voltage graph. In this case the obtained quantum graph is quantum isomorphic to a classical graph. As a main result we also prove a quantum version of the Gross-Tucker theorem which characterizes precisely which graphs can be written as derived graphs of voltage graphs.