On the Connectivity and the Diameter of Betweenness-Uniform Graphs

Abstract

Betweenness centrality is a centrality measure based on the overall amount of shortest paths passing through a given vertex. A graph is betweenness-uniform if all its vertices have the same betweenness centrality. We study the properties of betweenness-uniform graphs. In particular, we show that every connected betweenness-uniform graph is either a cycle or a 3-connected graph. Also, we show that betweenness uniform graphs of high maximal degree have small diameter.

Constructions of betweenness-uniform graphs from trees

“…[3] Actually, any betweenness-uniform graph is 3-connected, unless it is isomorphic to a cycle. [5] There exist other studies concerned with both the values of betweenness in betweenness-uniform graphs [6] and edge betweenness-uniform graphs [8].…”