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Topological Analysis of Local Structure in Atomic Systems
Published in Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef, Statistical Methods for Materials Science, 2019
Emanuel A. Lazar, David J. Srolovitz
The language and tools of graph theory can be used to record complete topological information of a Voronoi cell by looking at it as a planar graph. Briefly, a graph is a set of points called vertices, and a set of connections between those vertices are called edges. A planar graph is one whose vertices and edges can be drawn in the plane without any edges crossing. Two graphs are isomorphic if there is a correspondence between their vertices so that two vertices are connected by an edge in one graph if and only if corresponding vertices are connected by an edge in the other graph [1059]. Mathematical theorems from the early twentieth century [965, 1142] guarantee that every Voronoi cell can be uniquely represented as a planar graph, thus allowing us to make precise statements about Voronoi cells using the language of graph theory. Figure 15.8 illustrates planar graphs corresponding to the three Voronoi cells of Figure 15.7.
Graphs and Surfaces
Published in Kenneth H. Rosen, Graphs, Algorithms, and Optimization, 2005
We know from $$$xref rid="chapter12" ref-type="book-part">chapter 12%%%/xref$$$ that a connected planar graph with n vertices, ε edges, and f faces satisfies Euler’s formula n — ε + f = 2. Furthermore, the skeleton of a polyhedron is a planar graph, so that any polyhedral division of the sphere also satisfies this formula. We say that the Euler characteristic of the sphere is 2.
Preliminaries
Published in Christopher M. Gold, Spatial Context: An Introduction to Fundamental Computer Algorithms for Spatial Analysis, 2018
Euler’s formula for planar graphs states that V + F − E = 2, where V, F and E are the numbers of vertices, faces and edges respectively. The ‘exterior’ of the graph counts as a face – this is most easily seen if you draw your graph on a balloon! In Figure 40 this is: 5 + 5 – 8 = 2.
A topological characterisation of looped drainage networks
Published in Structure and Infrastructure Engineering, 2022
Didrik Meijer, Hans Korving, François Clemens-Meyer
“A plane graph is a graph drawn in the plane in such a way that any pair of edges meet only at their end vertices (if they meet at all). A planar graph is a graph which is isomorphic to a plane graph, i.e. it can be (re)drawn as a plane graph” (Clark & Holton, 1991, p. 157). In contrast to planar graphs, non-planar graphs can have intersection links (e.g. rail networks or flyovers, airline networks, cargo ship networks or the internet) (Barthélemy, 2011).