  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.
