We study two characteristics of a small-world graph proposed by Zhang et al. to model complex networks. Our study relies on the recursive structure of the graph. Firstly, we use it to design a labelling scheme in order to create an implicit routing (i.e., a routing scheme based on the label of vertices). Secondly, proving the average distance in this graph was arduous, thus Zhang et al. chose to study the diameter: we establish a closed-form formula of the average distance, proved using the recursive structure. Thus, we characterize that the graph is small-world and not ultra small-world as was still possible. Our proof is of particular interest for other graphs based on similar recursive structures.