Some Results on Bichon's Quantum Automorphism Group of Graphs

The notion of the quantum automorphism group of a graph was introduced by J. Bichon in 2003 and T. Banica in 2005 respectively. This article explores primarily the quantum automorphism group of a graph $\Gamma$, denoted by $QAut_{Bic}(\Gamma)$, in Bichon's framework. First, we provide a sufficient condition for non-commutativity of Bichon's quantum automorphism group and discuss several applications of this criterion. Although it is known that $QAut_{Bic}(\Gamma) \cong QAut_{Bic}(\Gamma^c)$ does not hold in general, we identify a family of graphs for which this isomorphism enforces that the graph has no quantum symmetry. Moreover, we describe a few families of graphs having quantum symmetries whose quantum automorphism groups in Bichon's sense are commutative. Finally, we show that free product, tensor product and free wreath product constructions can arise as Bichon's quantum automorphism groups of connected graphs in the following sense: For a finite family of compact matrix quantum groups $\{Q_i\}_{i=1}^{m}$ arising as Bichon's quantum automorphism groups of certain graphs, there exist connected graphs $\Gamma_{free}$, $\Gamma_{ten}$ and $\Gamma_{wr}$ whose quantum automorphism groups are $*_{i=1}^{m} Q_{i}$, $\otimes_{i=1}^{m} Q_i$ and $Q_1 \wr_{*} Q_2$ respectively.