Given rules for substituting the edges of a graph by subgraphs, it is of interest to determine the properties of the resulting iterative graphs. One of the significant problems in this respect is to determine the fractality dimension of the generated graphs. By considering the Hausdorff dimension and the Box Counting Dimension in Euclidean space, we are may by way of analogy also calculate the Box Counting Dimension for sequences of substituted graphs. To generalize our model further, we allow edges to be colored and let substitution rules depend on edge colours. In addition, we prove that scale-free effects are preserved through these graph iterations.