Sum of weighted distances in trees

Abstract

The sum of distances between vertices of a graph is a mathematical concept that has numerous direct applications in many different fields. The so-called extremal problem asks for the structures that maximize or minimize this sum of distances among a given category of graphs under certain constraints. Such extremal problems have been extensively studied, especially for various classes of trees. The same type of extremal questions for the very natural generalization to edge-weighted trees, however, has received surprisingly little attention. This is possibly due to the complexities of such questions.In this paper we consider trees with a given set of non-negative edge weights and study the extremal structures (and the extremal assignment of the weights) that maximize or minimize the sum of weighted distances between vertices. This is done through considering the extremal trees with given degree sequences and then comparing such extremal trees of different degree sequences and weight assignments.Although our arguments appear to be very technical, we explain the intuitive background by relating our results to those on unweighted sum of distances. Indeed, along with other consequences, it is easy to see that previously established extremal results on the sum of unweighted distances are simply the special cases when all edge weights are 1. Related interesting questions for potential future work are also mentioned.