Relationship between Edge-wiener Index and Gutman Index of a Graph

The Wiener indexW (G) of a connected graphG is defined to be the sum ∑ 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 ∑ 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 ∑ 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 We(G) ≥ 1 4 Gut(G) − 1 4 |E(G)| + 3 4 κ3(G) + 3κ4(G) where κm(G) denotes the number of all m-cliques in G. Moreover, equality holds if and only if G is a tree or a complete graph. Using this result we show that We(G) ≥ δ−1 4 W (G) where δ denotes the minimum degree in G.