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In the mathematical field of graph theory, a Platonic graph is a graph that has one of the Platonic solids as its skeleton. There are 5 Platonic graphs, and all of them are regular, polyhedral (and therefore by necessity also 3-vertex-connected, vertex-transitive, edge-transitive and planar graphs), and also Hamiltonian graphs.[1]

The platonic graphs can be seen as Schlegel diagrams of the platonic solids. (excluding the square pyramid also show here)

Tetrahedral graph – 4 vertices, 6 edges
Octahedral graph – 6 vertices, 12 edges
Cubical graph – 8 vertices, 12 edges
Icosahedral graph – 12 vertices, 30 edges
Dodecahedral graph – 20 vertices, 30 edges

Orthogonal projections of platonic solids

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