  We consider the problem of estimating the number of clusters ($k$) in a
dataset. We propose a non-parametric approach to the problem that is based on
maximizing a statistic constructed from similarity graphs. This graph-based
statistic is a robust summary measure of the similarity information among
observations and is applicable even if the number of dimensions or number of
clusters is possibly large. The approach is straightforward to implement,
computationally fast, and can be paired with any kind of clustering technique.
Asymptotic theory is developed to establish the selection consistency of the
proposed approach. Simulation studies demonstrate that the graph-based
statistic outperforms existing methods for estimating $k$. We illustrate its
utility on a high-dimensional image dataset and RNA-seq dataset.
