Wiener index in weighted graphs via unification of Θ∗Θ∗-classes

It is proved that the Wiener index of a weighted graph (G,w)(G,w) can be expressed as the sum of the Wiener indices of weighted quotient graphs with respect to an arbitrary combination of Θ∗Θ∗-classes. Here Θ∗Θ∗ denotes the transitive closure of Djoković–Winkler’s relation ΘΘ. 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.