Recent developments in quantum chemistry, perturbative quantum field theory,
statistical physics or stochastic differential equations require the
introduction of new families of Feynman-type diagrams. These new families arise
in various ways. In some generalizations of the classical diagrams, the notion
of Feynman propagator is extended to generalized propagators connecting more
than two vertices of the graphs. In some others (introduced in the present
article), the diagrams, associated to noncommuting product of operators inherit
from the noncommutativity of the products extra graphical properties. The
purpose of the present article is to introduce a general way of dealing with
such diagrams. We prove in particular a "universal" linked cluster theorem and
introduce, in the process, a Feynman-type "diagrammatics" that allows to handle
simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams
arising from the study of interacting systems (such as the ones where the
ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or
electric field, by impurities...) or Wightman fields (that is, expectation
values of products of interacting fields). Our diagrammatics seems to be the
first attempt to encode in a unified algebraic framework such a wide variety of
situations. In the process, we promote two ideas. First, Feynman-type
diagrammatics belong mathematically to the theory of linear forms on
combinatorial Hopf algebras. Second, linked cluster-type theorems rely
ultimately on M\"obius inversion on the partition lattice. The two theories
should therefore be introduced and presented accordingl
