Characterizing partition functions of the edge-coloring model by rank growth

We characterize which graph invariants are partition functions of an edge-coloring model over the complex numbers, in terms of the rank growth of associated `connection matrices'

Graph parameters from symplectic group invariants

In this paper we introduce, and characterize, a class of graph parameters obtained from tensor invariants of the symplectic group. These parameters are similar to partition functions of vertex models, as introduced by de la Harpe and Jones, [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]. Yet they give a completely different class of graph invariants. We moreover show that certain evaluations of the cycle partition polynomial, as defined by Martin [P. Martin, Enum\'erations eul\'eriennes dans les multigraphes et invariants de Tutte-Grothendieck, Diss. Institut National Polytechnique de Grenoble-INPG; Universit\'e Joseph-Fourier-Grenoble I, 1977], give examples of graph parameters that can be obtained this way.Comment: Some corrections have been made on the basis of referee comments. 21 pages, 1 figure. Accepted in JCT

On the exact learnability of graph parameters: The case of partition functions

We study the exact learnability of real valued graph parameters $f$ which are known to be representable as partition functions which count the number of weighted homomorphisms into a graph $H$ with vertex weights $\alpha$ and edge weights $\beta$. M. Freedman, L. Lov\'asz and A. Schrijver have given a characterization of these graph parameters in terms of the $k$-connection matrices $C(f,k)$ of $f$. Our model of learnability is based on D. Angluin's model of exact learning using membership and equivalence queries. Given such a graph parameter $f$, the learner can ask for the values of $f$ for graphs of their choice, and they can formulate hypotheses in terms of the connection matrices $C(f,k)$ of $f$. The teacher can accept the hypothesis as correct, or provide a counterexample consisting of a graph. Our main result shows that in this scenario, a very large class of partition functions, the rigid partition functions, can be learned in time polynomial in the size of $H$ and the size of the largest counterexample in the Blum-Shub-Smale model of computation over the reals with unit cost.Comment: 14 pages, full version of the MFCS 2016 conference pape

On traces of tensor representations of diagrams

Let $T$ be a set, of {\em types}, and let \iota,o:T\to\oZ_+. A {\em $T$-diagram} is a locally ordered directed graph $G$ equipped with a function $\tau:V(G)\to T$ such that each vertex $v$ of $G$ has indegree $\iota(\tau(v))$ and outdegree $o(\tau(v))$. (A directed graph is {\em locally ordered} if at each vertex $v$, linear orders of the edges entering $v$ and of the edges leaving $v$ are specified.) Let $V$ be a finite-dimensional \oF-linear space, where \oF is an algebraically closed field of characteristic 0. A function $R$ on $T$ assigning to each $t\in T$ a tensor $R(t)\in V^{*\otimes \iota(t)}\otimes V^{\otimes o(t)}$ is called a {\em tensor representation} of $T$. The {\em trace} (or {\em partition function}) of $R$ is the \oF-valued function $p_R$ on the collection of $T$-diagrams obtained by `decorating' each vertex $v$ of a $T$-diagram $G$ with the tensor $R(\tau(v))$, and contracting tensors along each edge of $G$, while respecting the order of the edges entering $v$ and leaving $v$. In this way we obtain a {\em tensor network}. We characterize which functions on $T$-diagrams are traces, and show that each trace comes from a unique `strongly nondegenerate' tensor representation. The theorem applies to virtual knot diagrams, chord diagrams, and group representations

Mixed partition functions and exponentially bounded edge-connection rank

We study graph parameters whose associated edge-connection matrices have exponentially bounded rank growth. Our main result is an explicit construction of a large class of graph parameters with this property that we call mixed partition functions. Mixed partition functions can be seen as a generalization of partition functions of vertex models, as introduced by de la Harpe and Jones, [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] and they are related to invariant theory of orthosymplectic supergroup. We moreover show that evaluations of the characteristic polynomial of a simple graph are examples of mixed partition functions, answering a question of de la Harpe and Jones.Comment: To appear in Ann. Inst. Henri Poincar\'e Comb. Phys. Interac

Tensor invariants for certain subgroups of the orthogonal group

Let V be an n-dimensional vector space and let On be the orthogonal group. Motivated by a question of B. Szegedy (B. Szegedy, Edge coloring models and reflection positivity, Journal of the American Mathematical Society Volume 20, Number 4, 2007), about the rank of edge connection matrices of partition functions of vertex models, we give a combinatorial parameterization of tensors in V \otimes k invariant under certain subgroups of the orthogonal group. This allows us to give an answer to this question for vertex models with values in an algebraically closed field of characteristic zero.Comment: 14 pages, figure. We fixed a few typo's. To appear in Journal of Algebraic Combinatoric