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The subject of this dissertation is the minimal AKLT model, a special, well-behaved isotropic model defined on any general graph. This model was first constructed in order to study the Heisenberg model which is conjectured by Haldane to have a unique gapped ground state when the spin is an integer. This model can be constructed on any arbitrary graph, and a natural question is which graphs can this model be shown to have a unique ground state with a stable spectral gap and which can be shown to have a degenerate or Neel ordered ground state. There are few such examples: the one-dimensional chain is known to have a unique stably gapped ground state, the hexagonal chain is known to have a unique gapped ground state, the hexagonal lattice is known to have a unique ground state with exponential decay of correlations, and a spectral gap . The Cayley trees of degree 2,3,4 are known to have unique ground states with exponential decay of correlations and those of degree d>4 are known to have a Neel ordered degenerate ground state space. Numerical evidence exists for the spectral gap of the square lattice. Beyond these not much is known. In this thesis we extend previous results in multiple directions to several infinite classes of graphs: we show that quasi-one-dimensional graphs that are bipartite have a unique ground state with exponential decay of correlations and a stable spectral gap. We show that under certain conditions, trees constructed from repeated bipartite graphs have degenerate, long-range ordered ground states, as well as irregular trees with a sufficiently high minimum splitting number. We also prove that the hexagonal lattice model has a stable spectral gap using techniques unrelated to the first two proofs. Each of these uses completely different styles of techniques which show the richness of the model. Lastly, in the conclusion, we provide some refined conjectures for the macroscopic behavior of this model on general lattices.