A type of centrality functions for a directed graph

Abstract

In an undirected graph obtained by abstracting traffic and communication networks, evaluation of the centrality of a vertex is interesting from the viewpoint of graph theory. the purpose of discussion of the centrality function is to provide a theoretical basis for solution of the problem. This paper is an attempt to extend the notion of the centrality function defined for the ujidirected graph to the directed graph. the existence conditions of the quasi‐stable and unstable graphs for the centrality functions with general coefficients are derived from the viewpoint of structural stability of the undirected graphs, for which the classification of the graphs is made by the shift of the center by adding an edge with the central vertex as an end point. Furthermore, the stability of the directed graphs is discussed.

On centrality functions of a graph

“…The definitions and the theorems discussed so far can be extended for a directed graph [6]. Let us begin with some preliminary definitions.…”