  We study the three state toric homogeneous Markov chain model and three
special cases of it, namely: (i) when the initial state parameters are
constant, (ii) without self-loops, and (iii) when both cases are satisfied at
the same time. Using as a key tool a directed multigraph associated to the
model, the state-graph, we give a bound on the number of vertices of the
polytope associated to the model which does not depend on the time. Based on
our computations, we also conjecture the stabilization of the f-vector of the
polytope, analyze the normality of the semigroup, give conjectural bounds on
the degree of the Markov bases.
