You maybe interested in my work with Terrence Bisson, we have studied closed models in various categories of graphs in particular in the category of directed graphs in which we have defined a closed model for which two finite graphs are homotopic equivalent if and only if they are isospectral.
We have defined another closed model in the category of directed graphs related to dynamic systems which characterizes the topology conjugacy class of some walkable finite graphs.
In a paper, I have defined a closed model structure which characterize strongly connected component. I have also studied closed models in the category of undirected graphs and shown that there does not exist a closed model which characterize the Ihara zeta function.
You maybe also interested in the work of Deborah Vicinsky who has constructed spectra for the model that I have defined with Bisson which characterizes isospectral graphs.
Symbolic dynamics and the category of graphs T Bisson, A Tsemo - Theory and Applications of Categories, 2011
Homotopy equivalence of isospectral graphs T Bisson, A Tsemo - New York J. Math, 2011
Closed models, strongly connected components and Euler graphs T Aristide - Proyecciones (Antofagasta), 2016
Tsemo Aristide
Applications of closed models defined by counting to graph theory and topology Algebra Letters, Vol 2017 (2017), Article ID 2
Vicinsky Deborah
The homotopy calculus of categories and graphs. https://scholarsbank.uoregon.edu/xmlui/bitstream/handle/1794/19283/Vicinsky_oregon_0171A_11298.pdf?sequence=1