Quantum field theory is surprisingly effective at describing various interesting algebraic-geometric and topological invariants. In this lecture series, we will examine this fascinating connection between physics and mathematics by focusing on simple topologies: graphs. The quantum field theory framework allows us to solve numerous interesting graph counting problems, many inaccessible using traditional methods. I will introduce this elegant combinatorial framework focusing on asymptotic graph enumeration. We will discuss applications in topology and, if time permits, statistical mechanics and complex networks.