The atom-bond connectivity index (ABC) is a molecular structure descriptor that recently found a remarkable application in rationalizing the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. Let G be a molecular graph, d(i) the degree of its vertex i, and ij the edge connecting the vertices i and j. Then ABC - Sigma(ij)root(d(i) + d(i) - 2)/(d(i)d(j)). Several properties of ABC are established. In particular, if a new edge is inserted into G, then ABC necessarily increases. By means of this result we characterize the graphs extremal w.r.t. ABC. (C) 2011 Elsevier B.V. All rights reserved.