In Chapter 5 of this thesis we characterize which graph parameters are partition functions of complex edge-coloring 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 vertex-coloring models are edge-reflection 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 edge-coloring 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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