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 Sigma(u,v) d(u, v) of the distances between the pairs of vertices in G. Similarly, the edge-Wiener index We(G) of G is defined to be the sum Sigma(e,f) d(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 Sigma(u,v) deg(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) >= 1/4Gut(G) - 1/4 vertical bar E(G)vertical bar + 3/4 kappa(3)(G) + 3 kappa(4)(G) where kappa(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) >= delta(2)-1/4W(G) where delta denotes the minimum degree in G. (C) 2014 Elsevier B.V. All rights reserved.