The US County Graph


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We took the publicly available polygon data of the 3141 US counties, threw them into Mathematica and made them into a graph, the US county graph. Two counties are connected if they share a common boundary point (this is not quite the official definition but was easier to compute). The graph has dimension 1047839931/501781280=2.08824 and Euler characteristic v-e+f-c+w=-5 where v=3141 is the number of vertices, e=9137 the number of edges, f the number of triangles, c the number of tetrahedra and w the number of K5 subgraphs. The graph is not planar (there are points, where 5 counties meet). Its close to a geometric graph but taking not positive length borders includes some tetrahedral and hypertetrahedra subgraphs which increases the dimension slightly. To the right is the vertex distribution:
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World Continent Graphs

Each vertex is a country. Two vertices are connected if they have a common border.
Africa America
Asia Europe

World