Wiener Index of Line Graphs

The Wiener index W(G) of a connected graph G is the sum of distances of all pairs of vertices in G. We show that for any connected graph G with delta(G) >= 2, W(L(G)) >= W(G), with equality if and only if G is a cycle, where L(G) is the line graph of G. We also present lower and upper hounds for W(L(G)) in terms of a graph invariant called Gutman index of G. As an immediate consequence, a relation between Gutman index and Wiener index of a tree is derived.