Planar graph of regular octahedron
Move the gray vertices of the octahedron on the left picture to get planar graph on the right.
Planar graph (Schlegel diagram) of a convex polyhedra lack scale, distance and shape, but the relationship between points is maintained.
Euler's formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely large region), then v - e + f = 2. Thanks to Schlegel diagram it is clear that Euler's formula is also valid for convex polyhedra.
The skeleton of the octahedron (the vertices and edges) form a graph. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.