  This paper presents a detailed review of both theory and algorithms for the
Cheeger cut based on the graph $1$-Laplacian. In virtue of the cell structure
of the feasible set, we propose a cell descend (CD) framework for achieving the
Cheeger cut. While plugging the relaxation to guarantee the decrease of the
objective value in the feasible set, from which both the inverse power (IP)
method and the steepest descent (SD) method can also be recovered, we are able
to get two specified CD methods. Comparisons of all these methods are conducted
on several typical graphs.

Spectral Theory