To find a qualitative answer we take the maximum This gives a qualitative result and highlights a few regions with long distances. What is the longest trajectory across the London underground? One way to answer this question is by looking at the distance matrix Millions of people traverse the London tube every day without realizing they do Graphs can be found everywhere in everyday life although it’s not always perceived as such. Say, we take a random graph of 5 vertices and 7 edges Assuming that the symmetry of this graph is the key to invariance is misleading thoughĪnother way to transform a graph into another one is by asking what it takes to turn the given graph into a complete one. It’s rather easy to create a simple case, the triangle graph is indeed topologically invariantĪnd this is true for any cycle graph. For example, a line graph of a given graph turns edges into vertices and vice versa īeing a transformation, one can wonder whether there are graphs which remain invariant under this type of transformation. Once you have a graph there are many ways in which it can be turned into another graph. Although a spanning tree is not unique it still gives a good indication of how many edges (shortcuts) break the tree-like structureĪ more numerical value can be obtained from the GraphDifference How far away from a tree? That’s something we can answer by looking at spanning trees. Turning this graph into a polyhedron is not possible (in a straightforward manner)but we can turn it into a 3d graph, which reveals that it’s almost a tree. The letter g will be used in this text to denote a graph. The vertices in this case are contained in two graphs, we can use the GraphUnion command to combine them and thus create a new graph. Polyhedrons can be combined to give beautiful graphics The results do not agree thoughīecause the vertex count of the polyhedron is based on its 3D wrapping while the unwrapped graph duplicates some vertices. The number of vertices can be found in two ways here, either using the VertexCount property of the PolyHedronData or through the VertexCount property of Graph. Polyhedral surfaces have been around since antiquity although it wasn’t recognized until Euler that they are in fact closely related to graphs. At the same time, various Mathematica symbols and techniques are presented which we will cover more in depth in later article. It demonstrates the ubiquity of graphs and how Mathematica makes it easy to experiment with ideas. This is an overview of the many ways in which one can create graphs in Mathematica.
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