  Linkage methods are among the most popular algorithms for hierarchical
clustering. Despite their relevance the current knowledge regarding the quality
of the clustering produced by these methods is limited. Here, we improve the
currently available bounds on the maximum diameter of the clustering obtained
by complete-link for metric spaces.
  One of our new bounds, in contrast to the existing ones, allows us to
separate complete-link from single-link in terms of approximation for the
diameter, which corroborates the common perception that the former is more
suitable than the latter when the goal is producing compact clusters.
  We also show that our techniques can be employed to derive upper bounds on
the cohesion of a class of linkage methods that includes the quite popular
average-link.
