After reading the paper On the Expressive Power of Homomorphism Count, I was wondering if there were some nice material I could read trying to study graphs from a categorical point of view. Stuff like trying to make it in an abelian category, looking at exact sequences of graphs (maybe you'd need to consider some kind of pointed graphs category), maybe defining stuff like Ext groups and studying them.
I hope my question makes sens.
Thanks in advance for any answer.