Average distance and generalised packing in graphs

Let G be a connected finite graph. The average distance μ ( G ) of G is the average of the distances between all pairs of vertices of G . For a positive integer k , a k -packing of G is a subset S of the vertex set of G such that the distance between any two vertices in S is greater than k . The k -packing number β k ( G ) of G is the maximum cardinality of a k -packing of G . We prove upper bounds on the average distance in terms of β k ( G ) and show that for fixed k the bounds are, up to an additive constant, best possible. As a corollary, we obtain an upper bound on the average distance in terms of the k -domination number, the smallest cardinality of a set S of vertices of G such that every vertex of G is within distance k of some vertex of S .