EXISTENCE OF QUANTUM SYMMETRIES FOR GRAPHS ON UP TO SEVEN VERTICES: A COMPUTER BASED APPROACH

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 classical symmetries, and it is a complex task to establish when this distinction occurs for a given graph, specifically assessing whether the algebra linked to the quantum automorphism group is commutative. We employ Gröbner base computations facilitated by the Gap and the Singular package Letterplace to address this problem. Our findings reveal the existence of quantum symmetries for all connected, undirected graphs without multiple edges or self-edges, comprising up to seven vertices. Notably, we conclude that if the classical automorphism group has order one or two, it serves as an obstruction to the existence of quantum symmetries.