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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.
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex.
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, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron, one of the Archimedean solids. It has 60 vertices and 120 edges, and is a quartic graph Archimedean graph .
[14] [15] Its graph called the octahedral graph, a Platonic graph. [4] The octahedral graph can be considered as complete tripartite graph,,, a graph partitioned into three independent sets each consisting of two opposite vertices. [16] More generally, it is a Turán graph,.
For degree two, any odd cycle is such a graph, and for degree three, four, and five, these graphs can be constructed from platonic solids by replacing a single edge by a path of two adjacent edges. In Vizing's planar graph conjecture , Vizing (1965) states that all simple, planar graphs with maximum degree six or seven are of class one, closing ...
It is also a 3-connected graph, meaning that, whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected. [27] [28] The skeleton of a cube can be represented as the graph, and it is called the cubical graph, a Platonic graph. It has the same number of vertices and edges as the cube, twelve ...
If G has C components (disconnected graphs), the same argument by induction on F shows that + = . One of the few graph theory papers of Cauchy also proves this result. Via stereographic projection the plane maps to the 2-sphere, such that a connected graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2.