Question

Definition: A polytope has property X iff there is a function f:N⁺ → R⁺ such that for each pair of vertices v_i, v_j the following holds:

dist_{euclideuclidean(vi, vj) = f(disttopologicalcombinatorial(vi, vj))}

with dist_{topologicalcombinatorial(vi, vj) = shortest path of edges between vi and vj.}

That means: for each v_i1, v_j1, v_i2, v_j2:

dist_{euclideuclidean(vi1, vj1) = disteuclideuclidean(vi2, vj2)}

iff

dist_{topologicalcombinatorial(vi1, vj1) = disttopologicalcombinatorial(vi2, vj2)}

Question 1: Is property X already named? What's its common name?

Question 2: Which polytopes have property X? The regular polytopes seem to have it, but are there more?