Applications of closed models defined by counting to graph theory and topology

In this paper, we define the notion of closed models defined by counting, and we compute their homotopy categories. We apply this construction to various categories of graphs. We show that there does not exist a closed model in the category of undirected graphs which characterizes the Ihara Zeta function in the sense that, a morphism $f:X\rightarrow Y$ is a weak equivalence for this model if and only if it induces a bijection between the sets of non degenerated cycles of $X$ and $Y$. Finally, we apply our construction to Galoisian complexes and dessins d'enfant.