Cut method and Djoković-Winklerʼs relation

Let Θ⁎ be the transitive closure of the Djoković-Winklerʼs relation Θ. 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. A related result for edge-weighted graphs is also given and a class of graphs studied in [H. Yousefi-Azari, M. H. Khalifeh, and A. R. Ashrafi, Calculating the edge Wiener and Szeged indices of graphs, J. Comput. Appl. Math. 235 (2011), 4866–4870] is characterized as partial cubes. We will Compute distance polynomial functions on graphs with transitive Djoković-Winklerʼs relation.