Wiener index versus Szeged index in networks

Let (G,w)(G,w) be a network, that is, a graph G=(V(G),E(G))G=(V(G),E(G)) together with the weight function w:E(G)→R+w:E(G)→R+. The Szeged index Sz(G,w)Sz(G,w) of the network (G,w)(G,w) is introduced and proved that Sz(G,w)≥W(G,w)Sz(G,w)≥W(G,w) holds for any connected network where W(G,w)W(G,w) is the Wiener index of (G,w)(G,w). Moreover, equality holds if and only if (G,w)(G,w) is a block network in which ww is constant on each of its blocks. Analogous result holds for vertex-weighted graphs as well.