Remoteness of graphs with given size and connectivity constraints

Let G be a finite, simple connected graph. The average distance of a vertex v of G is the arithmetic mean of the distances from v to all other vertices of G. The remoteness ρ(G) of G is the maximum of the average distances of the vertices of G. In this paper, we give sharp upper bounds on the remoteness of a graph of given order, connectivity and size. We also obtain corresponding bound s for 2-edge-connected and 3-edge-connected graphs, and bounds in terms of order and size for triangle-free graphs.