  We consider the class of spin Hamiltonians on a 1D chain with periodic
boundary conditions that are (i) translational invariant, (ii) commuting and
(iii) scale invariant, where by the latter we mean that the ground state
degeneracy is independent of the system size. We correspond a directed graph to
a Hamiltonian of this form and show that the structure of its ground space can
be read from the cycles of the graph. We show that the ground state degeneracy
is the only parameter that distinguishes the phases of these Hamiltonians. Our
main tool in this paper is the idea of Bravyi and Vyalyi (2005) in using the
representation theory of finite dimensional C^*-algebras to study commuting
Hamiltonians.
