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 μ c ( G ) = N 2 − 1 ∑ { u , v } ⊆ V c ( u ) c ( v ) d G ( u , v ) , where d G ( u , v ) denotes the usual distance between u and v in G . If c ( v ) = 1 for all vertices v of G , then μ c ( G ) is the ordinary average distance. sent sharp bounds on μ c 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 .