Solution group representations as quantum symmetries of graphs

TL;DR

This paper explores quantum symmetries of graphs derived from linear constraint systems, revealing new relationships between quantum automorphism groups and graph structures, including examples with quantum symmetry and isomorphism.

Contribution

It establishes the duality between quantum automorphism groups and solution groups, and constructs new graphs with specific quantum symmetry properties.

Findings

Quantum automorphism group equals the dual of the solution group.

Constructed graphs with quantum symmetry and finite quantum automorphism group.

Provided examples of quantum isomorphic, non-isomorphic graphs with no quantum symmetry.

Abstract

In 2019, Aterias et al. constructed pairs of quantum isomorphic, non-isomorphic graphs from linear constraint systems. This article deals with quantum automorphisms and quantum isomorphisms of colored versions of those graphs. We show that the quantum automorphism group of such a colored graph is the dual of the homogeneous solution group of the underlying linear constraint system. Given a vertex- and edge-colored graph with certain properties, we construct an uncolored graph that has the same quantum automorphism group as the colored graph we started with. Using those results, we obtain the first known example of a graph that has quantum symmetry and finite quantum automorphism group. Furthermore, we construct a pair of quantum isomorphic, non-isomorphic graphs that both have no quantum symmetry.