  Graphs are topological spaces that include broader objects than discretized
manifolds, making them interesting playgrounds for the study of quantum phases
not realized by symmetry breaking. In particular they are known to support
anyons of an even richer variety than the two-dimensional space. We explore
this possibility by building a class of frustration-free and gapped
Hamiltonians based on discrete abelian gauge groups. The resulting models have
a ground state degeneracy that can be either a topological invariant, an
extensive quantity or a mixture of the two. For two basis of the degenerate
ground states which are complementary in quantum theory, the entanglement
entropy is exactly computed. The result for one basis has a constant global
term, known as the topological entanglement entropy, implying long-range
entanglement. On the other hand, the topological entanglement entropy vanishes
in the result for the other basis. Comparisons are made with similar
occurrences in the toric code. We analyze excitations and identify anyon-like
excitations that account for the topological entanglement entropy. An analogy
between the ground states of this system and the $\theta$-vacuum for a $U(1)$
gauge theory on a circle is also drawn.
