Independence and average distance in graphs

Abstract

Let G = ( V, E) be a simple connected graph of order n ⩾ 2 and let k ⩾ 0 be an integer. A subset X ⊆ V is k-independent if the distance between every two vertices of X is at least k + 1. We define the k-independence number of G, I k ( G), to be the maximum cardinality among all k -independent sets of G. Best possible upper bounds are established for I k ( G), as functions of n and k, together with a lower bound which generalizes an earlier result for the case k = 1. We obtain sharp lower bounds for the average distance in terms of the k-independence number, and cite the extremal graphs.