Related papers: A generalized Kac-Ward formula

For a graph $G$, let $\sigma_{2}(G)$ be the minimum degree sum of two non-adjacent vertices in $G$. A chord of a cycle in a graph $G$ is an edge of $G$ joining two non-consecutive vertices of the cycle. In this paper, we prove the following…

In 1952, Kac and Ward developed a combinatorial formulation for the two dimensional Ising model which is another method of obtaining Onsager's formula for the free energy per site in the thermodynamic limit of the model. Feynman gave an…

The partition function for two-dimensional nearest neighbour Ising models in the presence of a magnetic field is derived . A comparison with the partition functions predicted by Onsager is carried out. The critical temperature estimated by…

We show that the n-fold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green…

The cage problem asks for the smallest number $c(k,g)$ of vertices in a $k$-regular graph of girth $g$ and graphs meeting this bound are known as cages. While cages are known to exist for all integers $k \ge 2$ and $g \ge 3$, the exact…

Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) =…

A general formula for the canonical partition function for a system obeying any statistics based on the permutation group is derived. The formula expresses the canonical partition function in terms of sums of Schur functions. The only…

We investigate the Kramers-Wannier approximation for the three-dimensional (3D) Ising model. The variational state is represented by an effective 2D Ising model, which contains two variational parameters. We numerically calculate the…

We apply localization techniques to compute the partition function of a two-dimensional N=(2,2) R-symmetric theory of vector and chiral multiplets on S^2. The path integral reduces to a sum over topological sectors of a matrix integral over…

We study the problem of partitioning the vertex set of a given graph so that each part induces a graph with components of bounded order; we are also interested in restricting these components to be paths. In particular, we say a graph $G$…

In this paper we propose an algorithm for the formation of matrices of isogeometric Galerkin methods. The algorithm is based on three ideas. The first is that we perform the external loop over the rows of the matrix. The second is that we…

MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given,…

We generalize the Fortuin-Kasteleyn (FK) cluster representation of the partition function of the Ising model to represent the partition function of quantum spin models with an arbitrary spin magnitude in arbitrary dimensions. This…

In this communication the advantages and drawbacks of the isogeometric analysis (IGA) are reviewed in the context of electromagnetic simulations. IGA extends the set of polynomial basis functions, commonly employed by the classical Finite…

Based on the results obtained in [Hucht, J. Phys. A: Math. Theor. 50, 065201 (2017)], we show that the partition function of the anisotropic square lattice Ising model on the $L \times M$ rectangle, with open boundary conditions in both…

We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a…

Graphical models represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function, is the main inference challenge relevant to multiple statistical and…

We compute the ${\cal N}=2$ supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the K\"ahler form with jumps…

Graph partitioning is the problem of dividing the nodes of a graph into balanced partitions while minimizing the edge cut across the partitions. Due to its combinatorial nature, many approximate solutions have been developed, including…

We suggest a generalization of the Feynman path integral to an integral over random surfaces. The proposed action is proportional to the linear size of the random surfaces and is called gonihedric. The convergence and the properties of the…