Proximity and Remoteness in Directed and Undirected Graphs

Abstract

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.

Proximity and remoteness in triangle-free and C_4-free graphs in terms of order and minimum degree

“…Clearly, g(X * ) = g(X) + 1, so X * beats X, a contradiction. This proves (1) . We now show that X = Z n,δ .…”

On the difference between proximity and other distance parameters in triangle-free graphs and $C_4$-free graphs

“…Such bounds for trees had been obtained by Sedlar [17]). For further results on proximity and remoteness, see for example [1,3,6,7,8,9,10,15,18].…”