A stability theorem for minimum edge graphs with given abstract automorphism group

Given a finite abstract group ?, whenever n is sufficiently large there exist graphs with n vertices and automorphism group isomorphic to 0. Let e (0, n) denote the minimum number of edges possible in such a graph. It is shown that for each ? there always exists a graph M such that for n sufficiently large, e(0, n) is attained by adding to M a standard maximal component asymmetric forest. A characterization of the graph M is given, a formula for e (0, n) is obtained (for large n), and the minimum edge problem is re-examined in the light of these results.