  The recently emerged spectral clustering surpasses conventional clustering
methods by detecting clusters of any shape without the convexity assumption.
Unfortunately, with a computational complexity of $O(n^3)$, it was infeasible
for multiple real applications, where $n$ could be large. This stimulates
researchers to propose the approximate spectral clustering (ASC). However, most
of ASC methods assumed that the number of clusters $k$ was known. In practice,
manual setting of $k$ could be subjective or time consuming. The proposed
algorithm has two relevance metrics for estimating $k$ in two vital steps of
ASC. One for selecting the eigenvectors spanning the embedding space, and the
other to discover the number of clusters in that space. The algorithm used a
growing neural gas (GNG) approximation, GNG is superior in preserving input
data topology. The experimental setup demonstrates the efficiency of the
proposed algorithm and its ability to compete with similar methods where $k$
was set manually.
