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Patterns in trees and quantum automorphism groups

Lucas Alger, Julie Capron, Félix de la Salle

Abstract

We prove that given a fixed finite tree $P$, almost all trees contain $P$ as a subtree. Moreover, the inclusion can be made so that it induces an embedding of the corresponding (quantum) automorphism groups, thereby providing generic properties of the latter.

Patterns in trees and quantum automorphism groups

Abstract

We prove that given a fixed finite tree , almost all trees contain as a subtree. Moreover, the inclusion can be made so that it induces an embedding of the corresponding (quantum) automorphism groups, thereby providing generic properties of the latter.
Paper Structure (4 sections, 8 theorems, 19 equations)

This paper contains 4 sections, 8 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The main result
  4. Application to quantum automorphism groups

Key Result

Lemma 2.5

Given the $p+1$ indices $j,i_1,...,i_p$, the number of rooted trees (j,T) with vertices $\{j,i_1,...,i_p\}$ isomorphic to $(v,P)$ is where $\mathrm{Aut}(v, P)$ denotes the subgroup of $\mathrm{Aut}(P)$ consisting in graph automorphisms fixing $v$.

Theorems & Definitions (20)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • ...and 10 more