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 @math -invariant tensors. Here @math is the sub- group of the real orthogonal group that stabilizes the vertex model. This answers a question of Bal 'azs Szegedy from 2007.
In @cite_2 , Schrijver introduced a different type of vertex connection matrix. It is also possible to characterize the rank of these connection matrices of partition functions of unweighted spin models (i.e. @math is the all ones vector) using our method. The characterization is similar to Theorem .
@cite_2
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Connection matrices were introduced in [M. Freedman, L. Lovasz, A. Schrijver, Reflection positivity, rank connectivity, and homomorphism of graphs (MSR Tech Report # MSR-TR-2004-41) ftp: ftp.research.microsoft.com pub tr TR-2004-41.pdf], where they were used to characterize graph homomorphism functions. The goal of this note is to determine the exact rank of these matrices. The result can be rephrased in terms of the dimension of graph algebras, also introduced in the same paper. Yet another version proves that if two k-tuples of nodes behave in the same way from the point of view of graph homomorphisms, then they are equivalent under the automorphism group. In this work, we show the deficiencies of the graph model for decomposing sparse matrices for parallel matrix-vector multiplication. Then, we propose two hypergraph models which avoid all deficiencies of the graph model. The proposed models reduce the decomposition problem to the well-known hypergraph partitioning problem widely encountered in circuit partitioning in VLSI. We have implemented fast Kernighan-Lin based graph and hypergraph partitioning heuristics and used the successful multilevel graph partitioning tool (Metis) for the experimental evaluation of the validity of the proposed hypergraph models. We have also developed a multilevel hypergraph partitioning heuristic for experimenting the performance of the multilevel approach on hypergraph partitioning. Experimental results on sparse matrices, selected from Harwell-Boeing collection and NETLIB suite, confirm both the validity of our proposed hypergraph models and appropriateness of the multilevel approach to hypergraph partitioning. In this paper we study the use of spectral techniques for graph partitioning. Let G = (V, E) be a graph whose vertex set has a ‘latent’ partition V1,. . ., Vk. Moreover, consider a ‘density matrix’ Ɛ = (Ɛvw)v, sw∈V such that, for v ∈ Vi and w ∈ Vj, the entry Ɛvw is the fraction of all possible Vi−Vj-edges that are actually present in G. We show that on input (G, k) the partition V1,. . ., Vk can (very nearly) be recovered in polynomial time via spectral methods, provided that the following holds: Ɛ approximates the adjacency matrix of G in the operator norm, for vertices v ∈ Vi, w ∈ Vj ≠ Vi the corresponding column vectors Ɛv, Ɛw are separated, and G is sufficiently ‘regular’ with respect to the matrix Ɛ. This result in particular applies to sparse graphs with bounded average degree as n = #V → ∞, and it has various consequences on partitioning random graphs. Abstract Spin models and vertex models on graphs are defined as appropriate generalizations of the Ising-Potts model of statistical mechanics. We review some of these state models and the graph functions defined by them. If a graph X represents a knot or a link L in R 3 , we describe models M for which the value Z M X at X of the graph function defined by M depends only on L and not on X .