This question is fundamentally a mathematical one, but it has a direct application to computer science.
I would like to build a network of n nodes (vertices) such that the distance between any two nodes, in terms of the number of steps between them, is a minimum. The network created will form a polyhedron, the only other parameters for which is that all nodes must have the same number of branches (edges).
I would like to compare two broad types of polyhedra. The first is a torus, which is formed by wrapping around a rhomboidal grid of hexagonal faces. The second is a Goldberg polyhedron (Class I, II, or III), which consists of precisely 12 pentagonal faces separated by some number of hexagonal faces. (See below.) The comparison is between polyhedra with the same number of vertices.
The geometric distances between any two vertices is of no consequence to my analysis since the network is virtual.
My questions are two: Does anyone know of a formula to determine the minimum number of edges joining any two vertices on a Goldberg polyhedron? And, if there are fewer edges between the most distant vertices on one polyhedron than another, would that necessarily imply that the average distance between any two vertices on that polyhedron will also be smaller?
A casual analysis suggests to me that the torus inherently provides a more direct route (having fewer edges) between any two points, but I am finding it difficult to picture the more complex Goldberg polyhedron.
Thank you in advance.