scholarly journals On Spanners of Geometric Graphs

2006 ◽  
pp. 388-399 ◽  
Author(s):  
Joachim Gudmundsson ◽  
Michiel Smid
Keyword(s):  
Geometric Graphs

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

Differential Equations on Networks (Geometric Graphs)

2004 ◽  
Vol 119 (6) ◽  
pp. 691-718 ◽  
Author(s):  
Yu. V. Pokornyi ◽  
A. V. Borovskikh
Keyword(s):  
Differential Equations ◽  
Geometric Graphs

scholarly journals On local transformations in plane geometric graphs embedded on small grids

2008 ◽  
Vol 39 (2) ◽  
pp. 65-77 ◽  
Author(s):  
Manuel Abellanas ◽  
Prosenjit Bose ◽  
Alfredo García ◽  
Ferran Hurtado ◽  
Pedro Ramos ◽  
...  
Keyword(s):  
Geometric Graphs

scholarly journals Convergence Theorems for Some Layout Measures on Random Lattice and Random Geometric Graphs

2000 ◽  
Vol 9 (6) ◽  
pp. 489-511 ◽  
Author(s):  
JOSEP DÍAZ ◽  
MATHEW D. PENROSE ◽  
JORDI PETIT ◽  
MARÍA SERNA
Keyword(s):  
Constant Factor ◽  
Geometric Graphs ◽  
Convergence Theorems ◽  
Linear Arrangement ◽  
Random Lattice ◽  
Random Geometric Graphs ◽  
Unit Square ◽  
Subcritical Regime ◽  
Probabilistic Setting ◽  
Lattice Graphs

This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection and Vertex Bisection. For full square lattices, we give optimal layouts for the problems still open. For arbitrary lattice graphs, we present best possible bounds disregarding a constant factor. We apply percolation theory to the study of lattice graphs in a probabilistic setting. In particular, we deal with the subcritical regime that this class of graphs exhibits and characterize the behaviour of several layout measures in this space of probability. We extend the results on random lattice graphs to random geometric graphs, which are graphs whose nodes are spread at random in the unit square and whose edges connect pairs of points which are within a given distance. We also characterize the behaviour of several layout measures on random geometric graphs in their subcritical regime. Our main results are convergence theorems that can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on random points in the unit square.

scholarly journals Symmetric motifs in random geometric graphs

2017 ◽  
Vol 6 (1) ◽  
pp. 95-105 ◽  
Author(s):  
Carl P Dettmann ◽  
Georgie Knight
Keyword(s):  
Geometric Graphs ◽  
Random Geometric Graphs

(Weakly) Self-approaching geometric graphs and spanners

2019 ◽  
Vol 78 ◽  
pp. 20-36 ◽  
Author(s):  
Davood Bakhshesh ◽  
Mohammad Farshi
Keyword(s):  
Geometric Graphs

scholarly journals Geometric Graphs with No Self-intersecting Path of Length Three

2002 ◽  
pp. 295-311 ◽  
Author(s):  
János Pach ◽  
Rom Pinchasi ◽  
Gábor Tardos ◽  
Géza Tóth
Keyword(s):  
Geometric Graphs

scholarly journals Clique Colourings of Geometric Graphs

10.37236/7159 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Colin McDiarmid ◽  
Dieter Mitsche ◽  
Pawel Prałat
Keyword(s):  
Asymptotic Behaviour ◽  
Chromatic Number ◽  
Euclidean Distance ◽  
High Probability ◽  
Maximal Clique ◽  
Higher Dimensions ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Vertex Set

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number.  Given $n$ points $\mathbf{x}_1, \ldots,\mathbf{x}_n$ in the plane, and a threshold $r>0$, the corresponding geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and $v_j$ are adjacent when the Euclidean distance between $\mathbf{x}_i$ and $\mathbf{x}_j$ is at most $r$. We investigate the clique chromatic number of such graphs.We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph $\mathcal{G}$ in the plane, where $n$ random points are independently and uniformly distributed in a suitable square. We see that as $r$ increases from 0, with high probability the clique chromatic number is 1 for very small $r$, then 2 for small $r$, then at least 3 for larger $r$, and finally drops back to 2.

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