> I once saw a man with a notebook and pencil drawing these kinds of diagrams, at the time I saw them as graph theory.

I have been engaged in some work on s-arc transitive graphs in algebraic graph theory. You'd be surprised how rarely I have to draw an actual graph. Most of the time my work involves reasoning about group actions, automorphisms, arc-stabilisers, etc.

For anyone curious what this looks like in practice, I have some brief notes here: <https://susam.net/26c.html#algebraic-graph-theory>. They do not cover the specific results on s-arc-transitivity I have been working on but they give a flavour of the area. A large part of graph theory proceeds without ever needing to draw specific graphs.