Some Results on Bichon's Quantum Automorphism Group of Graphs
Rajibul Haque, Ujjal Karmakar, Arnab Mandal
TL;DR
The paper advances the understanding of Bichon’s quantum automorphism groups QAut_{Bic}(Γ) by establishing a sharp non-commutativity criterion based on edge-free disjoint automorphisms, and by exploring how commutativity relates to Γ and Γ^c, including families with no quantum symmetry. It systematically analyzes how standard graph products influence QAut_{Bic} and demonstrates that, unlike Banica’s framework, parcels of these quantum groups can behave differently under complementation and products. The authors then develop three concrete constructions—free product, tensor, and free wreath product—that realize any finite family of CMQGs as QAut_{Bic} of connected graphs, using corona and related products to ensure connectedness and the desired quantum group structure. These results collectively expand the toolkit for engineering quantum symmetries of graphs and highlight structural differences between Bichon’s and Banica’s theories, with potential implications for quantum automorphism classifications and graph-product methodologies in CMQGs.
Abstract
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 $Γ$, denoted by $QAut_{Bic}(Γ)$, 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 $QAut_{Bic}(Γ) \cong QAut_{Bic}(Γ^c)$ 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 $Γ_{free}$, $Γ_{ten}$ and $Γ_{wr}$ whose quantum automorphism groups are $*_{i=1}^{m} Q_{i}$, $\otimes_{i=1}^{m} Q_i$ and $Q_1 \wr_{*} Q_2$ respectively.