On quantum Cayley graphs

TL;DR

This paper clarifies the relationship between two approaches to quantum graphs, extends the theory to infinite dimensions, and introduces quantum Cayley graphs associated with discrete quantum groups.

Contribution

It provides a unified framework connecting quantum adjacency matrices and quantum relations, and defines quantum Cayley graphs for discrete quantum groups.

Findings

Clarified the correspondence between quantum adjacency matrices and quantum relations.

Extended quantum graph theory to infinite-dimensional cases.

Introduced quantum Cayley graphs for discrete quantum groups.

Abstract

We clarify the correspondence between the two approaches to quantum graphs: via quantum adjacency matrices and via quantum relations. We show how the choice of a (possibly non-tracial) weight manifests itself on the quantum relation side and suggest an extension of the theory of quantum graphs to the infinite dimensional case. Then we use this framework to introduce quantum graphs associated to discrete quantum groups, leading to a new definition of a quantum Cayley graph.