This thesis is concerned with links between certain graph parameters and the invariant theory of the orthogonal group and
some of its subgroups. These links are given through socalled partition functions of edgecoloring models. These partition
functions can be seen as graph parameters as well as polynomials that are invariant under a natural action of the orthogonal
group. As graph parameters they may be seen as generalizations of counting the number of linegraph homomorphisms.
In Chapter 5 of this thesis we characterize which graph parameters are partition functions of complex edgecoloring models. This is done using the First and Second Fundamental Theorem from invariant theory and Hilbert’s Nulstellensatz. In Chapter 6 we give a combinatorial interpretation of algebras of tensors that are invariant under certain subgroups of the orthogonal group. Using some advanced techniques from geometric invariant theory, we characterize which partition functions of vertexcoloring models are edgereflection positive in Chapter 7. In Chapter 8 we prove a result on compact orbit spaces in Hilbert spaces and use this to develop a limit theory of edgecoloring models.
Our results are motivated by, and connected to, the rather recent field of graph limits and graph partition functions, whose study was initiated by Borgs, Chayes, Lovász, Schrijver, Sós, Szegedy and Vesztergombi.
In Chapter 5 of this thesis we characterize which graph parameters are partition functions of complex edgecoloring models. This is done using the First and Second Fundamental Theorem from invariant theory and Hilbert’s Nulstellensatz. In Chapter 6 we give a combinatorial interpretation of algebras of tensors that are invariant under certain subgroups of the orthogonal group. Using some advanced techniques from geometric invariant theory, we characterize which partition functions of vertexcoloring models are edgereflection positive in Chapter 7. In Chapter 8 we prove a result on compact orbit spaces in Hilbert spaces and use this to develop a limit theory of edgecoloring models.
Our results are motivated by, and connected to, the rather recent field of graph limits and graph partition functions, whose study was initiated by Borgs, Chayes, Lovász, Schrijver, Sós, Szegedy and Vesztergombi.
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Thesis

Cover

Title pages

Contents

Preface

1: Introduction

2: Preliminaries

3: Partition functions of edge and vertexcoloring models

4: Invariant theory

5: Characterizing partition functions of edgecoloring models

6: Connection matrices and algebras of invariant tensors

7: Edgereflection positive partition functions of vertexcoloring models

8: Compact orbit spaces in Hilbert spaces and limits of edgecoloring models

Summary

Samenvatting

Bibliography

Index

List of symbols

Errata

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