Proximity, remoteness and maximum degree in graphs

The average distance of a vertex v of a connected graph G is the arithmetic mean of the distances from v to all other vertices of G. The proximity pi(G) and the remoteness rho(G) of G are the minimum and the maximum of the average distances of the vertices of G, respectively. In this paper, we give upper bounds on the remoteness and proximity for graphs of given order, minimum degree and maximum degree. Our bounds are sharp apart from an additive constant.