On the Connectivity and the Diameter of Betweenness-Uniform Graphs

TL;DR

This paper investigates the structural properties of betweenness-uniform graphs, revealing that connected ones are either cycles or 3-connected, and those with high degree tend to have small diameters.

Contribution

It characterizes the structure of betweenness-uniform graphs and establishes bounds relating degree and diameter, advancing understanding of their topology.

Findings

Connected betweenness-uniform graphs are either cycles or 3-connected.

High-degree betweenness-uniform graphs have small diameters.

Structural properties of betweenness-uniform graphs are characterized.

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.