Quantum Symmetries of Vertex-Transitive Graphs on 12 Vertices

TL;DR

This paper investigates quantum symmetries of all vertex-transitive graphs on 12 vertices, expanding understanding of quantum automorphism groups and providing explicit results for most cases.

Contribution

It determines the quantum symmetries of all vertex-transitive graphs on 12 vertices, filling a gap in existing classifications.

Findings

Most 12-vertex vertex-transitive graphs have known quantum automorphism groups.

Explicit quantum automorphism groups are provided for the majority of these graphs.

The work extends previous classifications from 11 and 13 vertices to 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.