On centrality functions of a graph

Abstract

For a connected nondirected graph, a centrality function is a real valued function of the vertices defined as a linear combination of the numbers of the vertices classified according to the distance from a given vertex. Some fundamental properties of the centrality functions and the set of central vertices are summarized. Inserting an edge between a center and a vertex, the stability of the set of central vertices are investigated. For a weakly connected directed graph, we can prove similar theorems with respect to a generalized centrality function based on a new definition of the modified distance from a vertex to another vertex.