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

The Wiener index W(G)W(G) of a connected graph GG is defined to be the sum ∑u,vd(u,v)∑u,vd(u,v) of the distances between the pairs of vertices in GG. Similarly, the edge-Wiener index We(G)We(G) of GG is defined to be the sum ∑e,fd(e,f)∑e,fd(e,f) of the distances between the pairs of edges in GG, or equivalently, the Wiener index of the line graph L(G)L(G). Finally, the Gutman index Gut(G) is defined to be the sum ∑u,vdeg(u)deg(v)d(u,v)∑u,vdeg(u)deg(v)d(u,v), where deg(u)deg(u) denotes the degree of a vertex uu in GG. 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)≥14Gut(G)−14|E(G)|+34κ3(G)+3κ4(G) where κm(G)κm(G) denotes the number of all mm-cliques in GG. Moreover, the equality holds if and only if GG is a tree or a complete graph. Using this result we show that We(G)≥δ2−14W(G) where δδ denotes the minimum degree in GG.