Graph-theoretic properties of the Wiener index
On the same page there was a section: relation to chemical properties (http://en.wikipedia.org/wiki/Wiener_index#Relation_to_chemical_properties) Maybe references in that section mentions more about how this index can be used for graphs. Seems like you can estimate the overall structure of the graph if it is dense (low index) or sparse. Because the lengths of all shortest pairs would be shorter in a dense graph.