Characterizing Edge Betweenness-Uniform graphs

The betweenness centality of an edge e is, summed over all u, v ∈ V ( G ), the ratio of the number of shortest u, v -paths in G containing e to the number of shortest u, v -paths in G . Graphs whose vertices all have the same edge betweenness centrality are called edge betweeness-uniform . It was recently shown by Madaras, Hurajov´a, Newman, Mi-randa, Fl´orez, and Narayan that of the over 11.7 million graphs with ten vertices or fewer, only four graphs are edge betweenness-uniform but not edge-transitive. In this paper we present new results involving properties of betweenness-uniform graphs.