  We propose a new method for obtaining complete asymptotic expansions in a
systematic manner, which is suitable for counting sequences of various graph
families in dense regime. The core idea is to encode the two-dimensional array
of expansion coefficients into a special bivariate generating function, which
we call a coefficient generating function. We show that coefficient generating
functions possess certain general properties that make it possible to express
asymptotics in a short closed form. Also, in most scenarios, we indicate a
combinatorial meaning of the involved coefficients. Applications of our method
include asymptotics of connected graphs, irreducible tournaments, strongly
connected digraphs, satisfiable 2-SAT formulae and contradictory strongly
connected implication digraphs. Moreover, due to its flexibility, the method
allows to treat a wide range of structural variations, including fixing the
numbers of connected, irreducible, strongly connected and contradictory
components, having the variations on source-like, sink-like and isolated
components, or adding weights and marking variables.
