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

Björn Schäfer, Mariusz Tobolski

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.

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

Abstract

A voltage graph is a finite directed graph whose edges are labeled by elements of a finite group . A classical construction of Gross and Tucker associates to every voltage graph with vertex set a so-called derived graph with vertex set . 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.
Paper Structure (16 sections, 23 theorems, 156 equations, 5 figures)

This paper contains 16 sections, 23 theorems, 156 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgments
  4. Preliminaries
  5. Quantum sets
  6. Quantum graphs
  7. Definition of quantum isomorphisms
  8. Crossed products
  9. Crossed product quantum sets
  10. Landstad Duality
  11. Voltage and derived quantum graphs
  12. Properties of the derived quantum graph
  13. Connection to Gross and Tucker's construction
  14. Quantum isomorphisms
  15. Quantum Gross--Tucker Theorem
  16. ...and 1 more sections

Key Result

Theorem 1

If $\tilde{A} = (\tilde{A}_g)_{g \in G}$ is a voltage quantum graph on $(\tilde{B}, \tilde{\psi})$ with respect to the action $\hat{\alpha}$, then we have a quantum isomorphism where we denote by $\mathrm{triv}$ the trivial action of $\hat{G}$ on $(\tilde{B}, \tilde{\psi})$.

Figures (5)

  • Figure 1: The Petersen graph is the derived graph of a voltage graph over the group $\mathbb{Z}_5$.
  • Figure 2: The voltage graph $(\Gamma, \lambda)$ and its derived graph $\Gamma^\lambda$.
  • Figure 3: The graphs corresponding to the adjacency matrices $X_g$.
  • Figure 4: The graphs corresponding to the adjacency matrices $E_\rtimes^\ast \tilde{A}_g E_\rtimes$.
  • Figure 5: Voltage graphs leading to quantum graphs on $M_2(\mathbb{C})$

Theorems & Definitions (61)

  • Theorem 1: Theorem \ref{['qiso::thm:quantum_isomorphism_between_derived_graphs']}
  • Theorem 2: Theorem \ref{['gtt::thm:quantum_gross_tucker_theorem']}
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • Definition 2.8
  • ...and 51 more