scholarly journals Approximating the Edge Length of 2-Edge Connected Planar Geometric Graphs on a Set of Points

2012 ◽  
pp. 255-266 ◽  
Author(s):  
Stefan Dobrev ◽  
Evangelos Kranakis ◽  
Danny Krizanc ◽  
Oscar Morales-Ponce ◽  
Ladislav Stacho
Keyword(s):  
Edge Length ◽  
Geometric Graphs ◽  
Set Of Points

scholarly journals Almost sure central limit theorems in stochastic geometry

2020 ◽  
Vol 52 (3) ◽  
pp. 705-734
Author(s):  
Giovanni Luca Torrisi ◽  
Emilio Leonardi
Keyword(s):  
Central Limit ◽  
Stochastic Geometry ◽  
Limit Theorems ◽  
Voronoi Tessellation ◽  
Edge Length ◽  
Central Limit Theorems ◽  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
Poisson Space ◽  
Set Approximation

AbstractWe prove an almost sure central limit theorem on the Poisson space, which is perfectly tailored for stabilizing functionals arising in stochastic geometry. As a consequence, we provide almost sure central limit theorems for (i) the total edge length of the k-nearest neighbors random graph, (ii) the clique count in random geometric graphs, and (iii) the volume of the set approximation via the Poisson–Voronoi tessellation.

scholarly journals The Erdős-Sós conjecture for geometric graphs

2013 ◽  
Vol Vol. 15 no. 1 (Combinatorics) ◽  
Author(s):  
Luis Barba ◽  
Ruy Fabila-Monroy ◽  
Dolores Lara ◽  
Jesús Leaños ◽  
Cynthia Rodrıguez ◽  
...  
Keyword(s):  
Geometric Graph ◽  
Geometric Graphs ◽  
Convex Position ◽  
Minimum Number ◽  
International Audience ◽  
Set Of Points

Combinatorics International audience Let f(n,k) be the minimum number of edges that must be removed from some complete geometric graph G on n points, so that there exists a tree on k vertices that is no longer a planar subgraph of G. In this paper we show that ( 1 / 2 )n2 / k-1-n / 2≤f(n,k) ≤2 n(n-2) / k-2. For the case when k=n, we show that 2 ≤f(n,n) ≤3. For the case when k=n and G is a geometric graph on a set of points in convex position, we completely solve the problem and prove that at least three edges must be removed.

Video Graphics and High-Voltage Electron Microscopy For Analysis of Microtubule Distributions In Fixed Cells

1990 ◽  
Vol 48 (1) ◽  
pp. 524-525
Author(s):  
Richard Mcintosh ◽  
David Mastronarde ◽  
Kent McDonald ◽  
Rubai Ding
Keyword(s):  
Electron Microscopy ◽  
Cross Sections ◽  
Cell Shape ◽  
Video Camera ◽  
Computer Memory ◽  
High Voltage Electron Microscopy ◽  
Thin Sections ◽  
Voltage Electron ◽  
Morphogenetic Event ◽  
Set Of Points

Microtubules (MTs) are cytoplasmic polymers whose dynamics have an influence on cell shape and motility. MTs influence cell behavior both through their growth and disassembly and through the binding of enzymes to their surfaces. In either case, the positions of the MTs change over time as cells grow and develop. We are working on methods to determine where MTs are at different times during either the cell cycle or a morphogenetic event, using thin and thick sections for electron microscopy and computer graphics to model MT distributions.One approach is to track MTs through serial thin sections cut transverse to the MT axis. This work uses a video camera to digitize electron micrographs of cross sections through a MT system and create image files in computer memory. These are aligned and corrected for relative distortions by using the positions of 8 - 10 MTs on adjacent sections to define a general linear transformation that will align and warp adjacent images to an optimum fit. Two hundred MT images are then used to calculate an “average MT”, and this is cross-correlated with each micrograph in the serial set to locate points likely to correspond to MT centers. This set of points is refined through a discriminate analysis that explores each cross correlogram in the neighborhood of every point with a high correlation score.

Approach to spatial modeling of networks and groups of objects on the basis of the weighted graph construction procedure

2020 ◽  
Vol 964 (10) ◽  
pp. 49-58
Author(s):  
V.I. Bilan ◽  
A.N. Grigor’ev ◽  
G.G. Dmitrikov ◽  
E.A. Dudin
Keyword(s):  
Spatial Configuration ◽  
Weighted Graph ◽  
Edge Length ◽  
Buffer Zones ◽  
Typical Structure ◽  
Spatial Graph ◽  
Object Based ◽  
Complementary Graph ◽  
Spatial Parameters ◽  
The Given

The direction of research on the development of a scientific and methodological tool for the analysis of spatial objects in order to determine their generalized spatial parameters was selected. An approach to the problem of modeling networks and groups of objects based on the synthesis of a weighted graph is proposed. The spatial configuration of objects based on the given conditions is described by a weighted graph, the edge length of which is considered as the weight of the edges. A generalization to the typical structure of a spatial graph is formulated; its essence is representation of nodal elements as two-dimensional (polygonal) objects. To take into account the restrictions on the convergence of the vertices described by the buffer zones, a complementary graph is formed. An algorithm for constructing the implementation of a spatial object based on the sequential determination of vertices that comply with the given conditions is proposed. Using the software implementation of the developed algorithm, an experiment was performed to evaluate the spatial parameters of the simulated objects described by typical graph structures. The following parameters were investigated as spatial ones

scholarly journals Normalized Sombor Indices as Complexity Measures of Random Networks

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 976
Author(s):  
R. Aguilar-Sánchez ◽  
J. Méndez-Bermúdez ◽  
José Rodríguez ◽  
José Sigarreta
Keyword(s):  
Random Matrix ◽  
Adjacency Matrix ◽  
Matrix Theory ◽  
Computational Study ◽  
Random Networks ◽  
Geometric Graphs ◽  
Theory Approach ◽  
Complexity Measures ◽  
Random Geometric Graphs ◽  
Highly Correlated

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

scholarly journals Geometric graphs from data to aid classification tasks with Graph Convolutional Networks

Patterns ◽  
2021 ◽  
Vol 2 (4) ◽  
pp. 100237
Author(s):  
Yifan Qian ◽  
Paul Expert ◽  
Pietro Panzarasa ◽  
Mauricio Barahona
Keyword(s):  
Geometric Graphs ◽  
Convolutional Networks ◽  
Classification Tasks

scholarly journals Connectivity in one-dimensional soft random geometric graphs

2020 ◽  
Vol 102 (6) ◽  
Author(s):  
Michael Wilsher ◽  
Carl P. Dettmann ◽  
Ayalvadi Ganesh
Keyword(s):  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
One Dimensional

scholarly journals A plane quartic curve with twelve undulations

1945 ◽  
Vol 35 ◽  
pp. 10-13 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Homogeneous Coordinates ◽  
Quartic Curve ◽  
Base Points ◽  
Octahedral Group ◽  
Quartic Form ◽  
Image Position ◽  
Quartic Curves ◽  
Set Of Points

The pencil of quartic curveswhere x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.Since (1) may also be writtenthe four linesare, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.

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