Ratio of vertices to edges is always 2:3

Question

I noticed that the ratio of the vertices to edges on shapes with more than one vertice is always 2:3. Is there any equation or mathematical proof that backs this up?

For example, a cube has 8 vertices and 12 edges, and the ratio of vertices to edges is 8:12, which simplifies to 2:3.

Answer 1

Assuming your observation is about polyhedra, it is only true if the skeleton of the polyhedron is a $3$-regular graph, or alternatively if three edges meet at every vertex. So the octahedron doesn't fit, having $6$ vertices and $12$ edges.

The $2:3$ ratio for polyhedra with three edges meeting at every vertex is an easy consequence of the hand-shaking lemma.

Answer 2

Very easy to find examples and counter-examples:

Answer 3

An octoherdron has 6 vertexes and 12 edges.

But there is a rule that relates faces, edges, and vertices. Euler's (other) formula says that all simply connected polyhedra (i.e. no holes) have the same Euler characteristic. That is $F+V-E = 2$ For example, a cube has 6 face, 8 vertexes, and 12 edges.

A 1-holed polyhedral torus has $F+V-E = 0.$ Additional holes lower the Euler characteristic by 2.