  The classical Hausdorff dimension of finite or countable metric spaces is
zero. Recently, we defined a variant, called \emph{finite Hausdorff dimension},
which is not necessarily trivial on finite metric spaces. In this paper we
apply this to connected simple graphs, a class that provides many interesting
examples of finite metric spaces. There are two very different cases: one in
which the distance is coarse (and one is doing Graph Theory), and another case
in which the distance is much finer (and one is somewhere between graphs and
finite metric spaces).
