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This thesis provides an extension of the work of Dirk Kreimer and AlainConnes on the Hopf algebra structure of Feynman graphs and renormalization togeneral graphs. Additionally, an algebraic structure of the asymptotics offormal power series with factorial growth, which is compatible with the Hopfalgebraic structure, is introduced.  The Hopf algebraic structure on graphs permits the explicit enumeration ofgraphs with constraints for the allowed subgraphs. In the case of Feynmandiagrams a lattice structure, which will be introduced, exposes additionalunique properties for physical quantum field theories. The differential ring offactorially divergent power series allows the extraction of asymptotic resultsof implicitly defined power series with vanishing radius of convergence.Together both structures provide an algebraic formulation of large graphs withconstraints on the allowed subgraphs. These structures are motivated by andused to analyze renormalized zero-dimensional quantum field theory at highorders in perturbation theory.  As a pure application of the Hopf algebra structure, an Hopf algebraicinterpretation of the Legendre transformation in quantum field theory is given.The differential ring of factorially divergent power series will be used tosolve two asymptotic counting problems from combinatorics: The asymptoticnumber of connected chord diagrams and the number of simple permutations. Forboth asymptotic solutions, all order asymptotic expansions are provided asgenerating functions in closed form. Both structures are combined in anapplication to zero-dimensional quantum field theory. Various quantities areexplicitly given asymptotically in the zero-dimensional version of $\varphi^3$,$\varphi^4$, QED, quenched QED and Yukawa theory with their all orderasymptotic expansions.

Graphs in perturbation theory: Algebraic structure and asymptotics