Bichon’s Quantum Automorphism Group of Graphs Demonstrates Non-Commutativity and Symmetry Enforcement

The symmetries of networks, or graphs, are fundamental to understanding their properties, and mathematicians have long studied these symmetries through the lens of automorphism groups. Rajibul Haque, Ujjal Karmakar, and Arnab Mandal investigate a specific type of symmetry group known as Bichon’s quantum automorphism group, which offers a more refined way to characterise graph symmetries than traditional methods. Their work establishes a key condition determining when these quantum groups fail to commute, revealing a surprising lack of symmetry in certain graphs, and conversely identifies graph families where these groups behave predictably. Significantly, the researchers demonstrate that complex mathematical constructions, including free products and tensor products, can emerge directly from the automorphism groups of carefully constructed graphs, offering new connections between abstract algebra and the structure of networks.

Quantum Symmetries of Graphs and Products

This work systematically investigates the quantum symmetries of graphs, building upon Bichon’s framework and extending results to Banica’s more general quantum setting. Researchers developed a new criterion for determining when Bichon’s quantum automorphism group is non-commutative, introducing the concept of ‘edge-free disjoint automorphisms’ as a refinement of existing conditions. This criterion was then applied to demonstrate non-commutativity in a variety of graph families, including complete graphs and their complements, and line graphs, establishing that almost all trees exhibit non-commutative quantum symmetries. The study further explores how graph structure influences quantum symmetries, demonstrating that certain graph products, such as Cartesian, direct, and tensor products, preserve non-commutative properties, while the lexicographic product does not.

Researchers constructed infinitely many graphs possessing quantum symmetries while simultaneously exhibiting commutative Bichon’s and Banica’s quantum automorphism groups, challenging previous assumptions about the connection between these properties. This was achieved through careful selection of graphs satisfying specific eigenvalue-related conditions and leveraging graph products to create examples with desired symmetry characteristics. A significant achievement of this work lies in demonstrating the existence of connected graphs whose quantum symmetry groups correspond to free products, tensor products, and free wreath products of compact matrix quantum groups. Building upon existing results for trees, the team constructed connected graphs exhibiting these product structures within Bichon’s framework, a challenge not readily addressed by previous approaches. This construction involved utilizing the corona product to create graphs suitable for both Banica’s and Bichon’s settings, showcasing a methodological innovation in the field.

Non-Commutativity in Bichon’s Quantum Automorphism Groups

Scientists have rigorously investigated the quantum automorphism groups of graphs, specifically focusing on Bichon’s framework, and established key findings concerning their non-commutativity and structure. The research demonstrates a sufficient condition for determining when Bichon’s quantum automorphism group of a graph is non-commutative, achieved through the introduction of the concept of ‘edge-free disjoint automorphisms’. Applying this criterion, the team produced infinitely many graphs exhibiting non-commutative quantum automorphism groups in Bichon’s sense. The study reveals that, unlike in some other frameworks, the isomorphism between a graph and its complement does not generally hold; researchers identified graphs where their quantum automorphism groups are distinct.

Furthermore, the team demonstrated that both a graph and its line graph can simultaneously possess non-commutative Bichon’s quantum automorphism groups under specific conditions, and confirmed this holds for almost all trees. Investigations into complete graphs revealed necessary and sufficient conditions for their Bichon’s quantum automorphism groups to be non-commutative. The work also explores how graph products influence quantum automorphism groups, confirming that Cartesian, direct, and tensor products preserve certain properties within Bichon’s framework, while the lexicographic product does not. Notably, the team constructed infinitely many graphs possessing quantum symmetries while maintaining commutative Bichon’s quantum automorphism groups for both the graph and its complement, demonstrating a nuanced relationship between symmetry and group structure.

Graph Symmetries and Automorphism Group Structure

This research advances understanding of the automorphism groups of graphs, specifically within the framework of Bichon’s approach. Scientists have identified a sufficient condition for non-commutativity within these groups and demonstrated that certain graph families enforce a lack of symmetry. Importantly, the team describes scenarios where standard constructions, such as free products, tensor products, and free wreath products, can arise as Bichon’s automorphism groups of connected graphs, establishing a link between abstract algebraic structures and the symmetries inherent in graph structures. The work further clarifies the relationship between a graph and its complement, identifying conditions under which their automorphism groups differ. Researchers pinpointed specific properties, quantum symmetry, the presence of quadrangles in the complement, and commutativity of the Bichon automorphism group, that characterize a class of graphs exhibiting commutative Bichon automorphism groups. By constructing graphs satisfying these criteria, the team demonstrates the existence of an infinite family of graphs with this property, offering a foundation for further investigation into the algebraic properties of these structures.