Quantum automorphism groups of finite graphs

Abstract

Abstract. A quantum analogue of the automorphism group of a finite graph is introduced. These are quantum subgroups of the quantum permutation groups defined by Wang. The quantum automorphism group is a stronger invariant for finite graphs than the usual automorphism group. We get a quantum dihedral group D 4 .

A compositional approach to quantum functions

“…In Section 5, we quantise finite graphs and their homomorphisms and define the 2-category QGraph which encodes their compositional structure. We show that our definitions capture the quantum graph homomorphisms and isomorphisms of Mančinska and Roberson [38] and Atserias et al [5], as well as the finite-dimensional representation theory of Banica and Bichon's quantum automorphism group algebras [7,12]. We also show that quantum graph isomorphisms are precisely dagger-dualisable 1-morphisms in QGraph.…”

“…Quantum symmetry groups and noncommutative topology. The study of quantum permutation groups -quantum variants of the symmetric groups S n in noncommutative topology -was suggested by Connes, and carried out by Wang, Banica, Bichon and others [7,8,9,10,11,12,63]. These quantum permutation groups are compact quantum groups in the sense of Woronowicz [67], obtained from a universal construction [63].…”

The Morita Theory of Quantum Graph Isomorphisms

“…Quantum pseudo-telepathy is exhibited by graphs that are quantum but not classically isomorphic. This work builds on two recent articles, in which Lupini, Mančinska and Roberson [34] and the present authors [37] independently discovered a connection between these quantum isomorphisms and the quantum automorphism groups of graphs [6,9,10,11,14] studied in the framework of compact quantum groups [51]. This connection has already proven to be fruitful, introducing new quantum information-inspired techniques to the study of quantum automorphism groups [8,34].…”

“…For a quantum graph Γ, we write QAut(Γ) for the monoidal dagger category QGraphIso(Γ, Γ) of quantum automorphisms of Γ. For classical graphs Γ, the category QAut(Γ) (or rather the Hopf C * -algebra for which it is the category of finite-dimensional representations) has been studied in the context of compact quantum groups [6,9,10,11,12,14].…”

“…The automorphism groups of the complete graphs K 2 and K 3 have no nontrivial subgroups of central type, so by Corollary 5.4 cannot be part of a pseudo-telepathic graph pair. 14 The circulant graphs all have dihedral automorphism group, which acts on them as on any cycle graph. As with the cycle graph (Examples 4.4 and 4.17), there are up to two conjugacy classes of nontrivial central type subgroups (all isomorphic to Z 2 × Z 2 ), all of which have some trivial vertex stabilisers; so, by Corollary 5.4, they cannot be part of a pseudo-telepathic graph pair.…”