Average distance in weighted graphs

We consider the following generalisation of the average distance of a graph. Let G be a connected, finite graph with a nonnegative vertex weight function c. Let N be the total weight of the vertices. If N≠0,1, then the weighted average distance of G with respect to c is defined by ^μc(G)=N2^-1∑u,v⊆Vc(u)c(v)^dG(u,v), where ^dG(u,v) denotes the usual distance between u and v in G. If c(v)=1 for all vertices v of G, then ^μc(G) is the ordinary average distance. We present sharp bounds on ^μc for trees, cycles, and graphs with minimum degree at least 2. We show that some known results for the ordinary average distance also hold for the weighted average distance, provided that each vertex has weight at least 1.