Wiener index in weighted graphs via unification of -classes

It is proved that the Wiener index of a weighted graph ( G , w ) can be expressed as the sum of the Wiener indices of weighted quotient graphs with respect to an arbitrary combination of Θ ∗ -classes. Here Θ ∗ denotes the transitive closure of Djoković–Winkler’s relation Θ . A related result for edge-weighted graphs is also given and a class of graphs studied in Yousefi-Azari et al. (2011) [25] is characterized as partial cubes.