Quantum Symmetries of Vertex-Transitive Graphs on 12 Vertices

Abstract

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.

57 pages, including an appendix by Julien Schanz and Daniel Schultz