  Spectral algorithms are graph partitioning algorithms that partition a node
set of a graph into groups by using a spectral embedding map. Clustering
techniques based on the algorithms are referred to as spectral clustering and
are widely used in data analysis. To gain a better understanding of why
spectral clustering is successful, Peng et al. (2015) and Kolev and Mehlhorn
(2016) studied the behavior of a certain type of spectral algorithm for a class
of graphs, called well-clustered graphs. Specifically, they put an assumption
on graphs and showed the performance guarantee of the spectral algorithm under
it. The algorithm they studied used the spectral embedding map developed by Shi
and Malic (2000). In this paper, we improve on their results, giving a better
performance guarantee under a weaker assumption. We also evaluate the
performance of the spectral algorithm with the spectral embedding map developed
by Ng et al. (2001).
