One of the longstanding problems in spectral graph clustering (SGC) is the so-called model order selection problem: automated selection of the correct number of clusters. This is equivalent to the problem of finding the number of connected components or communities in an undirected graph. In this paper, we propose AMOS, an automated model order selection algorithm for SGC. Based on a recent analysis of clustering reliability for SGC under the random interconnection model, AMOS works by incrementally increasing the number of clusters, estimating the quality of identified clusters, and providing a series of clustering reliability tests. Consequently, AMOS outputs clusters of minimal model order with statistical clustering reliability guarantees. Comparing to three other automated graph clustering methods on real-world datasets, AMOS shows superior performance in terms of multiple external and internal clustering metrics. Our AMOS codes are available for download at https://github.com/tgensol/AMOS