A quantum Frucht's theorem and quantum automorphisms of quantum Cayley graphs
Michael Brannan, Daniel Gromada, Junichiro Matsuda, Adam Skalski, Mateusz Wasilewski
TL;DR
The article extends Frucht's theorem to the quantum setting by proving that every finite quantum group $\mathbb{G}$ is the quantum automorphism group of some undirected finite quantum graph, using quantum Cayley graphs as the fundamental building blocks. It develops a coloured-graph approach to decompose and then recombine symmetries, proving a quantum Frucht theorem via a two-step process: (i) construct a finite family of quantum Cayley graphs on $\mathbb{G}$ whose automorphism groups intersect to $\mathbb{G}$, and (ii) merge them into a single graph whose quantum automorphism group remains $\mathbb{G}$. The paper also proves dual-graph rigidity for non-abelian groups: there exists a quantum Cayley graph on $\widehat{\Gamma}$ with quantum automorphism group exactly $\widehat{\Gamma}$, and analyzes cases like duals of symmetric groups to illustrate the landscape and limitations of these constructions. Together, these results provide a systematic toolkit for realizing arbitrary finite quantum groups as quantum automorphism groups of quantum graphs and for producing examples with large gaps between classical and quantum symmetries.
Abstract
We establish a quantum version of Frucht's Theorem, proving that every finite quantum group is the quantum automorphism group of an undirected finite quantum graph. The construction is based on first considering several quantum Cayley graphs of the quantum group in question, and then providing a method to systematically combine them into a single quantum graph with the right symmetry properties. We also show that the dual $\widehat Γ$ of any non-abelian finite group $Γ$ is ``quantum rigid''. That is, $\widehat Γ$ always admits a quantum Cayley graph whose quantum automorphism group is exactly $\widehat Γ$.