Quantum symmetries of vertex-transitive graphs on 12 vertices

Recently, the work on quantum automorphism groups of graphs has seen renewed progress, which we expand in this paper. Quantum symmetry is a richer notion of symmetry than the classical symmetries of a graph. In general, it is non-trivial to decide whether a given graph does have quantum symmetries or not. For vertex-transitive graphs, the quantum symmetries have already been determined in earlier work on up to 11 and 13 vertices. This paper fills the gap by determining for all vertex-transitive graphs on 12 vertices, whether they have quantum symmetries and for most of these graphs, we also give their quantum automorphism group explicitly.