  The kth power of a simple graph G, denoted G^k, is the graph with vertex set
V(G) where two vertices are adjacent if they are within distance k in G. We are
interested in finding lower bounds on the average degree of G^k. Here we prove
that if G is connected with minimum degree d > 2 and |V(G)| > (8/3)d, then G^4
has average degree at least (7/3)d. We also prove that if G is a connected
d-regular graph on n vertices with diameter at least 3k+3, then the average
degree of G^{3k+2} is at least (2k+1)(d+1) - k(k+1) (d+1)^2/n - 1. Both of
these results are shown to be essentially best possible; the second is best
possible even when n/d is arbitrarily large.
