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In perturbative quantum field theory, the n-point functions consist, in general, of an infinity of Feynman graphs. Traditionally, these are generated via functional methods. This book describes the relation between complete, connected, and 1-particle irreducible n-point functions directly at the level of the Hopf algebra of time-ordered field operators. The ensembles of time-ordered n-point functions are simply linear forms on this algebra. It is showed, for instance, that the complete and connected n-point functions are elegantly related through the convolution product (induced by the coproduct). In this setting, a simple algebraic relation between connected and 1-particle irreducible n-point functions is derived, while the connected n-point functions are expressed in terms of their loop order contributions. At the center of the work stands a Hopf algebraic representation of graphs and a new algorithm to recursively generate all trees or all connected graphs and their values as Feynman graphs. This monograph presents a clear and self-contained exposition of all the results and their proofs. An introduction to the basic concepts required for the reading is also given.