Let D".", be the diameter of a connected undirected graph on n s= 2 vertices and n -1 ^ m ^ s{n) edges, where s(n) = n(n -l)/2. Then D",. V(;1) = 1, and for m < s(n) it is shown that 2 ^ D",", ^ n -[(V8(w -n) + 17 -1)/21. The bounds on D".", are sharp. Upper bound on Z> n?w . Since by definition of D n ", the graph is assumed to be connected, it follows that m^ n -1. Since D lus{n) = 1, we may assume that m < s(n).