Chemical structures of organic compounds are characterized numerically by a variety of structural descriptors computed either from the molecular graph or from the three-dimensional (3D) molecular geometry. Extensive use of such structural descriptors or topological indices has been made in drug design, screening of chemical databases, similarity and diversity assessment, and quantitative struct...

MOTIVATION Finding a good network null model for protein-protein interaction (PPI) networks is a fundamental issue. Such a model would provide insights into the interplay between network structure and biological function as well as into evolution. Also, network (graph) models are used to guide biological experiments and discover new biological features. It has been proposed that geometric rando...

Similar to Euclidean geometry, graph theory is a science that studies figures that consist of points and lines. The core of Euclidean geometry is the parallel postulate, which provides the basis of the geometric invariant that the sum of the angles in every triangle equals π and Cramer’s rule for solving simultaneous linear equations. Since the counterpart of parallel postulate in graph theory ...

A centrally symmetric convex body is a compact set with non-empty interior that about the origin. Of particular interest are those both smooth and strictly convex—known here as regular bodies—since they retain many of useful properties d-dimensional Euclidean ball. We prove for any given C, homothetic packing copies C randomly chosen radii will have (2, 2)-sparse planar contact graph. further t...

A grid graph is a node-induced finite subgraph of the infinite grid. It is rectangular if its set of nodes is the product of two intervals. Given a rectangular grid graph and two of its nodes, we give necessary and sufficient conditions for the graph to have a Hamilton path between these two nodes. In contrast, the Hamilton path (and circuit) problem for general grid graphs is shown to be NP-co...

Motivated by the problem of predicting the release kinetics of matrix tablets, we study random walks on the contact graph of a random sphere packing of spheres of two sizes. For a random walk on the unweighted graph that terminates in a specified target set, we compare the euclidean distance covered to the number of steps. We find a linear dependence of the former on the latter, with proportion...

Consider the following heuristic for planar Euclidean instances of the traveling salesman problem (TSP): select a subset of the edges which induces a planar graph, and solve either the TSP or its graphical relaxation on that graph. In this paper, we give several motivations for considering this heuristic, along with extensive computational results. It turns out that the Delaunay and greedy tria...

Computational geometry is concerned with the design and analysis of algorithms that solve problems on geometric data in R, where the dimension d is a constant. A large part of the field has been devoted to problems that involve distances determined by pairs of points in a given point set. Given a set S of n points in R, we may wish to compute a pair p, q of distinct points in S whose distance i...

For any given r ≥ 0, a random set of n points X1, . . . , Xn in the unit disc in the Euclidean plane induces a metric random graph G(n, r) with vertex set {1, . . . , n} and edge set { {i, j} : |Xi − Xj| ≤ r } where |Xi − Xj| denotes the Euclidean distance between the points Xi and Xj. Connectivity results analogous to the classical results of Erdös and Renyi forGn,p are shown to hold in this s...

All triangulations of euclidean oriented matroids are of the same PLhomeomorphism type by a result of Anderson. That means all triangulations of euclidean acyclic oriented matroids are PL-homeomorphic to PL-balls and that all triangulations of totally cyclic oriented matroids are PL-homeomorphic to PLspheres. For non-euclidean oriented matroids this question is wide open. One key point in the p...