Are there any textbooks on graph theory written for a category theorist?
It would probably have to be on directed graph theory, but if there's some trick we can use to talk about undirected graphs as well that would be interesting.
A little more specifically, I'm looking for a text that begins by defining directed graphs and paths, then defines the obvious category out of a given directed graph with paths as arrows, then proceeds to derive results about directed graphs using these categories.
Most connections I see made between category theory and graph theory are in the other direction, taking the underlying graph of a category and saying something about it to derive a result about the category, but as someone comfortable with categories and not comfortable with graphs this approach isn't particularly illuminating.
Further, unless I'm mistaken, these constructions amount to an equivalence (maybe even an isomorphism?) between the category of directed graphs and the category of categories, so it feels like we should be able to say something about directed graphs from this perspective.
Any references are appreciated.