A cyclic graph is a graph with at each vertex a cyclic order of the edges incident with it specied. We characterize which real-valued functions on the collection of cubic cyclic graphs are partition functions of a real vertex model (P. de la Harpe, V.F.R. Jones, Graph invariants related to statistical mechanical models: examples and problems, Journal of Combinatorial Theory, Series B 57 (1993) 207{227). They are characterized by `weak re ection positivity', which amounts to the positive semideniteness of matrices based on the `k-join' of cubic cyclic graphs (for all k 2 Z+). Basic tools are the representation theory of the symmetric group and geometric invariant theory, in particular the Hanlon-Wales theorem on the decomposition of Brauer algebras and the Procesi- Schwarz theorem on inequalities dening orbit spaces.

Journal of Combinatorial Theory - Series B
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Regts, G., Schrijver, L., & Sevenster, B. (2016). On partition functions of 3-graphs. Journal of Combinatorial Theory - Series B, 121, 421–431.