Almost all trees have quantum symmetry

TL;DR

This paper proves that, unlike most graphs, almost all trees possess quantum symmetry, extending classical symmetry results into the quantum domain using Banica's framework and quantum automorphism groups.

Contribution

It provides the first explicit proof that almost all trees have quantum symmetry, complementing classical results on tree symmetry and quantum graph automorphisms.

Findings

Almost all trees have quantum symmetry.

Presence of two cherries in a tree implies quantum symmetry.

Explicit proof of quantum symmetry for almost all trees.

Abstract

From the work of Erd\H{o}s and R\'{e}nyi from 1963 it is known that almost all graphs have no symmetry. In 2017, Lupini, Man\v{c}inska and Roberson proved a quantum counterpart: Almost all graphs have no quantum symmetry. Here, the notion of quantum symmetry is phrased in terms of Banica's definition of quantum automorphism groups of finite graphs from 2005, in the framework of Woronowicz's compact quantum groups. Now, Erd\H{o}s and R\'{e}nyi also proved a complementary result in 1963: Almost all trees do have symmetry. The crucial point is the almost sure existence of a cherry in a tree. But even more is true: We almost surely have two cherries in a tree - and we derive that almost all trees have quantum symmetry. We give an explicit proof of this quantum counterpart of Erd\H{o}s and R\'{e}nyi's result on trees.