  Approximate Spectral Clustering (ASC) is a popular and successful heuristic
for partitioning the nodes of a graph $G$ into clusters for which the ratio of
outside connections compared to the volume (sum of degrees) is small. ASC
consists of the following two subroutines: i) compute an approximate Spectral
Embedding via the Power method; and ii) partition the resulting vector set with
an approximate $k$-means clustering algorithm. The resulting $k$-means
partition naturally induces a $k$-way node partition of $G$.
  We give a comprehensive analysis of ASC building on the work of Peng et
al.~(SICOMP'17), Boutsidis et al.~(ICML'15) and Ostrovsky et al.~(JACM'13). We
show that ASC i) runs efficiently, and ii) yields a good approximation of an
optimal $k$-way node partition of $G$. Moreover, we strengthen the quality
guarantees of a structural result of Peng et al. by a factor of $k$, and
simultaneously weaken the eigenvalue gap assumption. Further, we show that ASC
finds a $k$-way node partition of $G$ with the strengthened quality guarantees.
