On the difference between the Szeged and the Wiener index

We prove a conjecture of Nadjafi-Arani et al. on the difference between the Szeged and the Wiener index of a graph (Nadjafi-Aranifi et al., 2012). Namely, if G is a 2-connected non-complete graph on n vertices, then Sz(G)−W(G)≥2n−6. Furthermore, the equality is obtained if and only if G is the complete graph Kn−1 with an extra vertex attached to either 2 or n−2 vertices of Kn−1. We apply our method to strengthen some known results on the difference between the Szeged and the Wiener index of bipartite graphs, graphs of girth at least five, and the difference between the revised Szeged and the Wiener index. We also propose a stronger version of the aforementioned conjecture.