The edge-Wiener index of a graph

If G is a connected graph, then the distance between two edges is, by definition, the distance between the corresponding vertices of the line graph of G . The edge-Wiener index W e of G is then equal to the sum of distances between all pairs of edges of G . We give bounds on W e in terms of order and size. In particular we prove the asymptotically sharp upper bound W e ( G ) ≤ 2 5 5 5 n 5 + O ( n 9 / 2 ) for graphs of order n .