On the sizes of graphs and their powers: The undirected case

Let GG be an undirected graph and GrGr be its rr-th power. We study different issues dealing with the number of edges in GG and GrGr. In particular, we answer the following question: given an integer r≥2r≥2 and all connected graphs GG of order nn such that Gr≠KnGr≠Kn, what is the minimum number of edges that are added when going from GG to GrGr, and which are the graphs achieving this bound?