My recommendation, try Lando and Zvonkin (2004): Graphs on Surfaces and Their Applications.

I think it is a great book which applies graphs embedded on surfaces to solving problems from other fields of mathematics. The style is very refreshing, vivid, and lively, I would say. The style reminded me of Hatcher's chapter 0 in his Algebraic Topology text, and of Matousek's book "Using the Borsuk-Ulam Theorem".

I would think the target audience of this book is graduate and research level, for some topics the pace is high. Exellent list of references, I think over 300.

**Edit:** I was just thinking, maybe the following quote from this book gives you a good indication. The authors are talking about a topological graph here:

"It is not merely a topological object, a graph *embedded into* (or *drawn on*) a two-dimensional surface. It is also a sequence of permutations (or, if you prefer, it "is encoded by" a sequence of permutations), which provides a relation to group theory. And it is at the same time a way of representing a ramified covering of the sphere by a compact two-dimensional manifold. Considering the sphere as the Riemann complex sphere we obtain, on the covering manifold, the structure of a Riemann surface. And Riemann surfaces rarely walk by themselves. Usually they keep company with Galois theory, with algebraic curves, moduli spaces and many other exciting subjects."