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