  The data clustering problem consists in dividing a data set into prescribed
groups of homogeneous data. This is a NP-hard problem that can be relaxed in
the spectral graph theory, where the optimal cuts of a graph are related to the
eigenvalues of graph $1$-Laplacian.
  In this paper, we give a first step for proving that the $k$-th Cheeger
constant is the minimum of the $1$-Laplacian Raylegh quotient among vectors
that are pseudo-orthogonal to the vectors realizing the previous $k-1$ Cheeger
constants. Moreover, we give new notations to punctually describe the paths,
among critical eigenvectors of the graph $1$-Laplacian, realizing sets with
prescribed genus.
  Eventually, we apply these results to give a method and a numerical
algorithm, based on a generalized inverse power method, to compute the third
Cheeger constant as the minimum of the $1$-Laplacian Raylegh quotient among
vectors orthogonal to the first and the second eigenvector of graph
$1$-Laplacian.
