Platonic graph
graph with a Platonic solid as its skeleton
octahedral graph
polyhedral graph of the octahedron, with 6 vertices and 12 edges
dodecahedral graph
polyhedral graph of the dodecahedron, with 20 vertices and 30 edges
tetrahedral graph
complete graph on 4 vertices
cubical graph
graph with 8 vertices and 12 edges
icosahedral graph
polyhedral graph of the icosahedron, with 12 vertices and 30 edges