14 May 2024: Peter Zemen, Technical University of Denmark.

14 May 2024: Peter Zemen, Technical University of Denmark. Quantum automorphism groups of finite graphs.

This talk will take place in the blue seminar room, back building Žitná 25, and on Zoom.

In this talk we introduce quantum automorphism groups of finite graphs. We motivate this concept from the point of view of quantum information theory and non-commutative geometry.

We give a characterisation of quantum automorphism groups of trees. In particular, for every tree, we show how to iteratively construct its quantum automorphism group using free products and free wreath products. This can be considered a quantum version of Jordan’s theorem for the automorphism groups of trees. This is one of the first characterisations of quantum automorphism groups of a natural class of graphs with quantum symmetry.

Sabidussi’s theorem gives necessary and sufficient conditions under which the automorphism group of a lexicographic product of two graphs is a wreath product of the respective automorphism groups. As a second result, we present a quantum version of Sabidussi’s theorem for finite graphs.

A common theme in both of our results is the use of Weisfeiler–Leman refinement procedure that is used to study classical automorphism groups of graphs.