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.