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 contravariant functor $D:C\rightarrow Set$ where 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 graphs is the $n$-cycle directed graph.
I have constructed with Bisson a Quillen model on the category $Gph$ such that a morphism $f:X\rightarrow Y$ is a weak equivalene if and only if for every $n>0$ the induced map $Hom(C_n,X)\rightarrow Hom(C_n,Y)$ 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:
Two finite directed graphs for the Quillen model above are weakly equivalent.
Two finite directed graphs have the same Zeta serie.
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 $\mathbb{Z}$-sets.
This shows that a finite graph is equivalent to finite set of cycles and enables to relate the degree of the factors of the characteristic polynomial of the finite graph $X$ to its image in the homotopy category which is a set of cycles.
For example, for this model, acyclic object are trees, and the characteristic polynomial of a tree is of the form $X^n$. In fact in this model if $X^p$ is a non trivial factor of the characteristic polynomial of the graph $G$, there is a weak cofibration $c:G'\rightarrow G$ (a whiskering) which is defined by attaching trees to $G'$.