Um ... bit of both. My goal is personal enjoyment - I've done graduate level maths [way in the past] but graph theory never really featured for some reason. Thanks for your continued consideration.
There are a lot of books out there for really applied and really theoretical versions. I personally prefer the CS perspective, which spends a lot of time on problems about graphs (finding matchings, covers, cliques, flows, traversals satisfying certain properties). A good overview text is Bondy & Murty[1]. This text sort of straddles theory and practice in that you get lots of topological intuition (like Euler characteristic), and lots of algorithms and applications. One thing that's conspicuously missing is a treatment of random graphs, which is a huge topic both in theory and in applications. I do like that the book covers some basic Ramsey theory, as this is one of the most popular topics in combinatorics and it gives you a good flavor of what's going on in modern research there.
