Average distance in weighted graphs

Let G be a given connected graph on n vertices. Suppose N facilities are located in the vertices of the graph. We consider the expected distance between two randomly chosen facilities. This is modelled by the following definition: Let G be a connected graph and let each vertex have weight c(v). The average distance of G with respect to c is defined as µ c (G) = N 2 −1 {u,v}⊆V (G) d G (u, v), where N = v∈V (G) c(v), and d G (u, v) is the distance in G between u and v, i.e., the length of a shortest u − v path in G. In this talk we consider bounds on µ c (G) in terms of properties of G. We specifically consider the case when G is a tree.