  We describe the structure of the graphs with the smallest average distance
and the largest average clustering given their order and size. There is usually
a unique graph with the largest average clustering, which at the same time has
the smallest possible average distance. In contrast, there are many graphs with
the same minimum average distance, ignoring their average clustering. The form
of these graphs is shown with analytical arguments. Finally, we measure the
sensitivity to rewiring of this architecture with respect to the clustering
coefficient, and we devise a method to make these networks more robust with
respect to vertex removal.
