Partition distance in graphs

If G is a graph and (mathcal{P}) is a partition of V(G), then the partition distance of G is the sum of the distance between all pairs of vertices that lie in the same part of (mathcal{P}). This concept generalizes several metric concepts and is dual to the concept of the colored distance due to Dankelmann, Goddard, and Slater. It is proved that the partition distance of a graph can be obtained from the Wiener index of weighted quotient graphs induced by the transitive closure of the Djokovi?–Winkler relation as well as by any partition coarser than it. It is demonstrated that earlier results follow from the obtained theorems. Applying the main results, upper bounds on the partition distance of trees with prescribed order and radius are proved and corresponding extremal trees characterized.