  The behavior of a certain random growth process is analyzed on arbitrary
regular and non-regular graphs. Our argument is based on the Expander Mixing
Lemma, which entails that the results are strongest for Ramanujan graphs, which
asymptotically maximize the spectral gap. Further, we consider
Erd\H{o}s--R\'enyi random graphs and compare our theoretical results with
computational experiments on flip graphs of point configurations. The latter is
relevant for enumerating triangulations.
