You may be interested with my work with Bisson, we have defined various closed models on category of graphs. In particular in the category of directed graphs. Consider the category $C$ which has two objects $0,1$ and two morphisms: $s,t:0\rightarrow 1$. A directed graph $G$ is a presheaf defined on $C$, that is two sets $G_0$ and $G_1$ where $G_0$ is the set of vertices here and $G_1$ the set of arrows $G(s):G_1\rightarrow G_0$ is the source map and $G(t)$ the target maps.
The category of undirected graphs can also be viewed as a topos, by adding in $C$ an involution $i:1\rightarrow 1$ such that $i\circ s=t$.
Bisson, Terrence, and Aristide Tsemo. "A homotopical algebra of graphs related to zeta series." Homology, Homotopy and Applications 11.1 (2009): 171-184.
Bisson, T., & Tsemo, A. (2011). Symbolic dynamics and the category of graphs. arXiv preprint arXiv:1104.1805.