We present a detailed study of the combinatorial interpretation of matrix integrals, including the examples of tessellations of arbitrary genera, and loop models on random surfaces. After reviewing their methods of solution, we apply these to the study of various folding problems arising from physics, including: the meander (or polymer folding) problem “enumeration of all topologically inequiva...

This paper gives lower bounds on the spectral radius of vertex-transitive graphs, based on the number of “prime cycles” at a vertex. The bounds are obtained by constructing circuits in the graph that resemble “cactus trees”, and enumerating them. Counting these circuits gives a coefficient-wise underestimation of the Green function of the graph, and hence an underestimation of its spectral radi...

In this paper, we investigate relationships between the volumes of cells of three-dimensional Voronoi tessellations and the lengths and areas of sections obtained by intersecting the tessellation with a randomly oriented plane. Here, in order to obtain analytical results, Voronoi cells are approximated to spheres. First, the probability density function for the lengths of the radii of the secti...

Voronoi tessellations specify a partition of a given space according to a set of sites where all points in that space are assigned to the closest site. Capacity-constrained Voronoi tessellations are special cases of Voronoi tessellations in which the region of each site has a predefined area. For example, the capacity constraint could state that each region in a Voronoi tessellation must have t...

A centroidal Voronoi tessellation is a Voronoi tessellation whose generating points are the centroids (centers of mass) of the corresponding Voronoi regions. We give some applications of such tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the territorial behavior of animals. We discuss methods f...

Abstract We introduce a class of systems Hamilton-Jacobi equations characterizing geodesic centroidal tessellations, i.e., tessellations domains with respect to distances where generators and centroids coincide. Typical examples are given by Voronoi power diagrams. An appropriate version the Fast Marching method on unstructured grids allows computing solution system and, therefore, associated t...