  The Cheeger inequalities give an upper and lower bound on the spectral gap of
discrete Laplacians defined on a graph in terms of the geometric
characteristics of the graph. We generalise this approach and we employ it to
determine if a given discrete Hamiltonian with non-positive elements is gapped
or not in the thermodynamic limit. First, we define the graph that corresponds
to such a generic Hamiltonian. Then we present a suitable generalisation of the
Cheeger inequalities that overcomes scaling deficiencies of the original
version. By employing simple examples we illustrate how the generalised Cheeger
inequalities can successfully identify gapped or gapless phases and we comment
on the computational complexity of this approach.
