A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting, the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.

The study of graph ${{C}^{*}}$-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph ${{C}^{*}}$-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph ${{C}^{*}}$-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.