No abstract

This paper defines anew the maximum two‐route flow as a measure to represent the relation between two vertices in the flow network. The maximum two‐route flow corresponds to the maximum number of communication channels that are composed of two‐route paths between two terminals in a communication network. The ℱ‐rings that can be composed simultaneously, containing the considered two vertices, is defined as the maximum two‐route flow between those two vertices. It is shown in this paper that a theorem exists for the maximum two‐route flow similar to the max‐flow min‐cut theorem for the ordinary flow.

This paper considers the problem of constructing a flow network when the centrality at each vertex is given. Here, attention is given to a centrality function such that the centrality at a vertex under consideration is the sum of maximum flow values between it and all other vertices. It is a representative centrality function among those representing the centrality of each vertex in an undirected flow network in which edges have capacity. First, we introduce a necessary and sufficient condition wherein a given sequence is a centrality sequence for a flow network. Next, we present a procedure for determining a terminal capacity matrix of a flow network with a given centrality sequence. Furthermore, we determine a terminal capacity matrix of a flow network with a given centrality sequence such that the sum of edge capacity is minimum. From these discussions we can see that the problem of constructing a flow network when a centrality is given, can be reduced to the previously known problem of constructing a flow network when a terminal capacity matrix is given.

It is of interest in graph theory to discuss the degree of centrality of a vertex in a graph or to locate the most central point in the graph representation of systems with network structures such as communication and traffic networks. Various expressions for the centrality of a point in a graph or a network have been considered in association with the above problem and are called centrality functions in general. Most of them are based on the distance between two points, while an expression for the centrality by the gauge between two points in a network is also proposed. Most of the previous centrality functions were defined by distance, whereas in this paper we consider the centrality function defined by the minimum cut separating two points in a network (or the maximum flow between two points). According to this centrality function, the centrality of the point under consideration may happen to be the sum of the maximum flows from the point to every other point in the special case and hence it seems to be significant in estimating traffic networks and reliability with respect to circuit faults in communication networks.

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.