Graph-theoretic properties of the Wiener index The Wiener index of a graph is the sum of the lengths of the shortest paths between all pairs of its vertices.

Are there useful graph-theoretic properties of this 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.