Can you assign twelve distinct positive integers, one to each edge of a cube, so that the sum of the three numbers assigned to the edges forming a vertex is always the same, and the total sum of the assigned numbers is minimized?

For referencing purposes, let the cube have vertices with coordinates: A(0,0,0), B(1,0,0), C(1,1,0), D(0,1,0), E(0,0,1), F(1,0,1), G(1,1,1), and H(0,1,1), so that the edges are: AB, BC, CD, DA, EF, FG, GH, HE, AE, BF, CG, and DH.