Number and weights of Feynman diagrams

The functional techniques of field theory are adapted to the problem of evaluating sums of combinatoric and group-theoretic weights of Feynman diagrams in ${\phi }^N$, quantum electrodynamics and non-Abelian theories. Considered are various classes of diagrams such as connected, one-particle-irreducible, and skeleton diagrams. For finite orders exact sums are given by compact recursion formulas. For higher orders estimates are obtained from the exact results or by steepest-descent methods.