What you are describing is [here][1]. A (directed) graph is essentially a **free category**, ie  the **path category** generated by the graph.

 This construction is the left adjoint to the forgetful functor from  Cats to the category of directed graphs. 

This amount to an equivalence between the category of directed graphs and the category of free categories, *not* the entire Cat (which makes sense, a category can be presented from a free cat by imposing relations, ie commuting diagrams, aside the trivial ones. That side is invisible from the point of view of the underlying graph). 

Now, on to the book: as far as I know, there isn't (though there are a few refs to the above, again look it up in the hyperlink). 
Such a book should investigate basic results of directed graph theory from the point of view of the theory of  free categories. Not too sure it would help finding new facts in graph theory, but it is nevertheless  an intriguing idea. After all, there is an entire industry on free groups, why not on free cats? 


  [1]: https://ncatlab.org/nlab/show/free+category