  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 firstly give new
notations to describe the paths, among critical eigenvectors of the graph
$1$-Laplacian, realizing sets with prescribed genus.
  We introduce the pseudo-orthogonality to characterize $m_3(G)$, a special
eigenvalue for the graph $1$-Laplacian. Furthermore, we use it to give an upper
bound for the third graph Cheeger constant $h_3(G)$, that is $h_3(G) \le
m_3(G)$. This is 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.
  Eventually, we apply these results to give a method and a numerical algorithm
to compute $m_3(G)$, based on a generalized inverse power method.
