Related papers: Dense packing on uniform lattices

We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that…

We investigate both analytically and by numerical simulation the kinetics of a microscopic model of hard rods adsorbing on a linear substrate, a model which is relevant for compaction of granular materials. The computer simulations use an…

We prove that for every positive integer $r$ and for every graph class $\mathcal G$ of bounded expansion, the $r$-Dominating Set problem admits a linear kernel on graphs from $\mathcal G$. Moreover, when $\mathcal G$ is only assumed to be…

A variety of real-world tasks involve the classification of images into pre-determined categories. Designing image classification algorithms that exhibit robustness to acquisition noise and image distortions, particularly when the available…

We revisit the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random with replacement. These matrices have a one-to-one correspondence with the…

The concept of hyperuniformity has been a useful tool in the study of large-scale density fluctuations in systems ranging across the natural and mathematical sciences. One can rank a large class of hyperuniform systems by their ability to…

The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhoods cover all vertices of $G$, while the packing number $\rho(G)$ is the maximum size of a vertex set whose closed neighborhoods are…

This paper deals with shape irregularity issues in discrete topology optimization algorithms whereby the design is created using the automated distribution of material in the design region. Graph theory is employed to derive appropriate…

We investigate the structure of large uniform random maps with $n$ edges, $\mathrm{f}_n$ faces, and with genus $\mathrm{g}_n$ in the so-called sparse case, where the ratio between the number vertices and edges tends to $1$. We focus on two…

We present the first systematic algorithm to estimate the maximum packing density of spheres when the grain sizes are drawn from an arbitrary size distribution. With an Apollonian filling rule, we implement our technique for disks in 2d and…

A simple lattice model is used to study compaction in granular media. As in real experiments, we consider a series of taps separated by large enough waiting times. The relaxation of the density exhibits the characteristic inverse…

A spanning subgraph of a graph $G$ is called a perfect star packing in $G$ if every component of the spanning subgraph is isomorphic to the star graph $K_{1,3}$. An efficient dominating set of graph $G$ is a vertex subset $D$ of $G$ such…

In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in $R^2$. With the benefit of hindsight, we show that the problem can be restricted to the…

Graph neural networks can accurately predict the chemical properties of many molecular systems, but their suitability for large, macromolecular assemblies such as gels is unknown. Here, graph neural networks were trained and optimised for…

We study the problems of counting copies and induced copies of a small pattern graph $H$ in a large host graph $G$. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns $H$. In…

Let $G$ be a simple graph with a perfect matching. Deng and Zhang showed that the maximum anti-forcing number of $G$ is no more than the cyclomatic number. In this paper, we get a novel upper bound on the maximum anti-forcing number of $G$…

We introduce and study the problem of balanced districting, where given an undirected graph with vertices carrying two types of weights (different population, resource types, etc) the goal is to maximize the total weights covered in vertex…

A powerful framework for studying graphs is to consider them as geometric graphs: nodes are randomly sampled from an underlying metric space, and any pair of nodes is connected if their distance is less than a specified neighborhood radius.…

We review various combinatorial applications of field theoretical and matrix model approaches to equilibrium statistical physics involving the enumeration of fixed and random lattice model configurations. We show how the structures of the…

Graph rigidity theory studies the capability of a graph embedded in the Euclidean space to constrain its global geometric shape via local constraints among nodes and edges, and has been widely exploited in network localization and formation…