A finite Hausdorff dimension for graphs

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).