Existence of quantum symmetries for graphs on up to seven vertices: a computer based approach

TL;DR

This paper investigates quantum symmetries of small graphs using computational algebra, determining when quantum automorphism groups are non-commutative for graphs with up to seven vertices.

Contribution

It provides a computational method to identify quantum symmetries in small graphs and establishes conditions under which classical automorphism groups obstruct quantum symmetries.

Findings

Quantum symmetries exist for certain small graphs.

Classical automorphism groups of order one or two hinder quantum symmetries.

A computational framework was developed using GAP and SINGULAR.

Abstract

The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the algebra associated to the quantum automorphism group is commutative. We use Gr\"obner base computations in order to tackle this problem; the implementation uses GAP and the SINGULAR package LETTERPLACE. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.