An Analogy in Geometric Homology: Rigidity and Cofactors on Geometric Graphs

Recent work on the crystallisation of inorganic crystals under compressed monomolecular surfactant films has shown that two dimensional templates can be used to promote the oriented nucleation of solids. When a suitable long alkyl chain surfactant is cast on the crystallisation media a monodispersied population of crystals forms exclusively at the monolayer/solution interface. Each crystal is aligned with a specific crystallographic axis perpendicular to the plane of the monolayer suggesting that nucleation is facilitated by recognition events between the nascent inorganic solid and the organic template.For example, monolayers of the long alkyl chain surfactant, stearic acid will promote the oriented nucleation of the calcium carbonate polymorph, calcite, on the (100) face, whereas compressed monolayers of n-eicosyl sulphate will induce calcite nucleation on the (001) face, (Figure 1 & 2). An extensive program of research has confirmed the general principle that molecular recognition events at the interface (including electrostatic interactions, geometric homology, stereochemical complementarity) can be used to promote the crystal engineering process.

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Normalized Sombor Indices as Complexity Measures of Random Networks

Entropy ◽

10.3390/e23080976 ◽

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.

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

Patterns ◽

10.1016/j.patter.2021.100237 ◽

2021 ◽

Vol 2 (4) ◽

pp. 100237

Author(s):

Yifan Qian ◽

Paul Expert ◽

Pietro Panzarasa ◽

Mauricio Barahona

Keyword(s):

Geometric Graphs ◽

Convolutional Networks ◽

Classification Tasks

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Connectivity in one-dimensional soft random geometric graphs

Physical Review E ◽

10.1103/physreve.102.062312 ◽

2020 ◽

Vol 102 (6) ◽

Author(s):

Michael Wilsher ◽

Carl P. Dettmann ◽

Ayalvadi Ganesh

Keyword(s):

Geometric Graphs ◽

Random Geometric Graphs ◽

One Dimensional

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Differential Equations on Networks (Geometric Graphs)

Journal of Mathematical Sciences ◽

10.1023/b:joth.0000012752.77290.fa ◽

2004 ◽

Vol 119 (6) ◽

pp. 691-718 ◽

Cited By ~ 47

Author(s):

Yu. V. Pokornyi ◽

A. V. Borovskikh

Keyword(s):

Differential Equations ◽

Geometric Graphs

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On local transformations in plane geometric graphs embedded on small grids

Computational Geometry ◽

10.1016/j.comgeo.2006.12.004 ◽

2008 ◽

Vol 39 (2) ◽

pp. 65-77 ◽

Cited By ~ 1

Author(s):

Manuel Abellanas ◽

Prosenjit Bose ◽

Alfredo García ◽

Ferran Hurtado ◽

Pedro Ramos ◽

...

Keyword(s):

Geometric Graphs

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

Combinatorics Probability Computing ◽

10.1017/s0963548300004454 ◽

2000 ◽

Vol 9 (6) ◽

pp. 489-511 ◽

Cited By ~ 27

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.

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