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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 π(G) and the remoteness ρ(G) of G are the minimum and the maximum of the average distances of the vertices of G. In this paper, we give upper bounds on the difference between the remoteness and proximity, the diameter and proximity, and the radius and proximity of a triangle-free graph with given order and minimum degree. We derive the latter two results by first proving lower bounds on the proximity in terms of order, minimum degree and either diameter or radius. Our bounds are sharp apart from an additive constant. We also obtain corresponding bounds for C 4 -free graphs.

Let D be a strongly connected digraph. The average distance σ(v) of a vertex v of D is the arithmetic mean of the distances from v to all other vertices of D. The remoteness ρ(D) and proximity π(D) of D are the maximum and the minimum of the average distances of the vertices of D, respectively. We obtain sharp upper and lower bounds on π(D) and ρ(D) as a function of the order n of D and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament T , we have π(T ) = ρ(T ) if and only if T is regular. Due to this result, one may conjecture that every strong digraph D with π(D) = ρ(D) is regular. We present an infinite family of non-regular strong digraphs D such that π(D) = ρ(D). We describe such a family for undirected graphs as well.

Let G be a finite, 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) and the proximity π(G) of G are the maximum and the minimum of the average distances of the vertices of G. In this paper, we present a sharp upper bound on the remoteness of a triangle-free graph of given order and minimum degree, and a corresponding bound on the proximity, which is sharp apart from an additive constant. We also present upper bounds on the remoteness and proximity of C 4 -free graphs of given order and minimum degree, and we demonstrate that these are close to being best possible.