  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.
