On a relation between Szeged and Wiener indices of bipartite graphs

Abstract

Hansen et‎. ‎al.‎, ‎using the AutoGraphiX software‎ ‎package‎, ‎conjectured that the Szeged index $Sz(G)$ and the‎ ‎Wiener index $W(G)$ of a connected bipartite graph $G$ with $n geq‎ ‎4$ vertices and $m geq n$ edges‎, ‎obeys the relation‎ ‎$Sz(G)-W(G) geq 4n-8$‎. ‎Moreover‎, ‎this bound would be the best possible‎. ‎This paper offers a proof to this conjecture‎.