The quantum automorphism group of a graph is a noncommutative generalisation of the classical automorphism group of a graph. Indeed, it can be viewed as a quantum group encoding the quantum symmetries of a graph. Explicitly determining the quantum automorphism group of a graph is a difficult task. In fact, it is computationally undecidable to even determine if a graph has quantum symmetry, i.e. if its quantum automorphism group is different from its automorphism group. 

 

In this talk, we shall see how a mixture of techniques from algebraic combinatorics and quantum groups can be used to express quantum automorphism groups of graphs as products of quantum automorphism groups of simpler graphs. If time permits, we shall also see a new product of quantum groups, and how it can be utilised to determine quantum automorphism groups of complicated substitutions of graphs. These techniques enable us to explicitly determine the quantum automorphism groups of large classes of graphs, such as trees and outerplanar graphs, and to describe quantum automorphism groups of disjoint unions  and lexicographic products of graphs.

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