Consider the category $Gph$ whose objects are directed graphs. That is the topos defined on the small category $C$ which has two objects $0,1$ and two morphisms $s,t:0\rightarrow 1$. A directed graph is thus defined by by a functor $D:C\rightarrow Set$, we call $D(0)$ the set of nodes and $D(1)$ the set of arrows, $D(s)$ is the source map and $D(t)$ is the target map. An example of graph is the $n$-cycle directe graph.

I have constructed with Bisson a Quillen model on the category $Gph$ such that two graphs such tha a weak equivalence $f:X\rightarrow Y$ for every $n>0$, the morphism $Hom(C_n,X)\rightarrow Hom(C_n,Y)$ is bijective.

Let $X$ is and $Y$ be finite graphs, we say that they are almost  isospectral if the characteristic polynomial of their adjecency matrices have the same non zero eigenvalues. 

The following conditions are equivalent:

1. Two finite directed  graphs for the Quillen model above are weakly equivalent.

2. Two finite directed graphs have the same Zeta serie.

3. Two finite directed graphs are almost isospectral.

The third condition implies that two finite graphs are weakly equivalent for the Quillen model if the factors of their characteristic polynomial distinct of $X^p$ are equal.

We have also computed the homotopy category of this closed model and showed that it is equivalent to the category of periodic $\mathbb{Z}$-set.

https://arxiv.org/pdf/0802.3859.pdf

https://arxiv.org/pdf/0906.4087.pdf