The rank of edge connection matrices and the dimension of algebras of invariant tensors

We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of G-invariant tensors. Here G is the subgroup of the real orthogonal group that stabilizes the vertex model. This answers a question of Balázs Szegedy from 2007

Approximate counting using Taylor’s theorem:a survey

In this article we consider certain well-known polynomials associated with graphs including the independence polynomial and the chromatic polynomial. These polynomials count certain objects in graphs: independent sets in the case of the independence polynomial and proper colourings in the case of the chro- matic polynomial. They also have interpretations as partition functions in statistical physics.The algorithmic problem of (approximately) computing these types of polyno- mials has been studied for close to 50 years, especially using Markov chain tech- niques. Around eight years ago, Barvinok devised a new algorithmic approach based on Taylor’s theorem for computing the permanent of certain matrices, and the approach has been applied to various graph polynomials since then. This arti- cle is intended as a gentle introduction to the approach as well as a partial survey of associated techniques and results