Relationship between the edge-Wiener index and the Gutman index of a graph

The Wiener index W(G) of a connected graph G is defined to be the sum @?"u","vd(u,v) of the distances between the pairs of vertices in G. Similarly, the edge-Wiener index W"e(G) of G is defined to be the sum @?"e","fd(e,f) of the distances between the pairs of edges in G, or equivalently, the Wiener index of the line graph L(G). Finally, the Gutman index Gut(G) is defined to be the sum @?"u","vdeg(u)deg(v)d(u,v), where deg(u) denotes the degree of a vertex u in G. In this paper we prove an inequality involving the edge-Wiener index and the Gutman index of a connected graph. In particular, we prove that W"e(G)>=14Gut(G)-14|E(G)|+34@k"3(G)+3@k"4(G) where @k"m(G) denotes the number of all m-cliques in G. Moreover, the equality holds if and only if G is a tree or a complete graph. Using this result we show that W"e(G)>=@d^2-14W(G) where @d denotes the minimum degree in G.