  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 on 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.
