The k-nucleus of a graph

In an earlier paper the measures of centrality in a graph G demonstrated by the center and centroid were unified and generalized, resulting in the definition of the h-centrum of G and denoted C(G;h) for 1 ≤ h ≤ |V(G)|. When dealing with the h-centrum one sums the distances from a given vertex u to each of the h vertices farthest from it. Here another way of generalizing the concepts of center and centroid is examined, and the k-nucleus is defined for G and denoted θk(G). When dealing with the k-nucleus one sums the distances from a given vertex u to the balls of radius k around each of the other vertices. The centroid of G is θo(G), and the center of G is θr(G), where r is the radius of G. Some relationships between the h-centra and the k-nuclei are presented.