In an asymmetric graph, do all nodes have distinct eigenvector centrality?

Question

I'm interested in small networks such that all nodes have distinct centrality. Asymmetry is necessary but not sufficient in the case of betweenness centrality (all peripheral nodes have zero betweenness centrality). I conjecture that eigenvector centralities must be distinct if there are no symmetries. Is that correct? How about other centrality measures?

Answer 1

In a $k$-regular graph, the Perron eigenvalue is $k$, and the eigenvector centralities are all equal. There are regular graphs with no symmetries, e.g. the Frucht graph: