Edge growth in graph powers

For a graph G, its rth power G is defined as the graph with the same vertex set as G, and an edge between any two vertices whenever they are within distance r of each other in G. Motivated by a result from additive number theory, Hegarty raised the question of how many new edges G has when G is a regular, connected graph with diameter at least r. We address this question for r = 3, 6. We give a lower bound for the number of edges in the rth power of G in terms of the order of G and the minimal degree of G. As a corollary, for r = 3, 6, we determine how small the ratio e(G)/e(G) can be for regular, connected graphs of diameter at least r.