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The polyhedral graph formed as the Schlegel diagram of a regular dodecahedron. In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected, planar graphs.
It can also be found in nature and supramolecules, as well as the shape of the universe. The skeleton of a regular dodecahedron can be represented as the graph called the dodecahedral graph, a Platonic graph. Its property of the Hamiltonian, a path visits all of its vertices exactly once, can be found in a toy called icosian game.
In the mathematical field of graph theory, an Archimedean graph is a graph that forms the skeleton of one of the Archimedean solids. There are 13 Archimedean graphs, and all of them are regular , polyhedral (and therefore by necessity also 3-vertex-connected planar graphs ), and also Hamiltonian graphs .
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The graph of a regular octahedron The skeleton of a regular octahedron can be represented as a graph according to Steinitz's theorem , provided the graph is planar —its edges of a graph are connected to every vertex without crossing other edges—and 3-connected graph —its edges remain connected whenever two of more three vertices of a ...
The web graph W 4,2 is a cube. The web graph W n,r is a graph consisting of r concentric copies of the cycle graph C n, with corresponding vertices connected by "spokes". Thus W n,1 is the same graph as C n, and W n,2 is a prism. A web graph has also been defined as a prism graph Y n+1, 3, with the edges of the outer cycle removed. [7] [10]
Hamiltonian platonic graphs: Image title: Orthographic projections and planar graphs of Hamiltonian cycles of the vertices of the five Platonic solids by CMG Lee. Only the octahedron has an Eulerian path, made by extending the Hamiltonian path with the dotted path. Width: 100%: Height: 100%