  Bott--Cattaneo's theory defines the integral invariants for a framed rational
homology 3-sphere equipped with an acyclic orthogonal local system, in terms of
graph cocycles without self-loops. The 2-loop term of their invariants is
associated with the theta graph. Their definition requires a cohomological
condition. Cattaneo--Shimizu removed this cohomological condition and gave a
2-loop invariant associated with a linear combination of the theta graph and
the dumbbell graph, the 2-loop trivalent graph with self-loops. In this paper,
we are concerned with an acyclic local system given by the adjoint
representation of a semi-simple Lie group composed with a representation of the
fundamental group of a closed 3-manifold, and we show that through a
cohomological construction eventually the integral associated with the dumbbell
graph vanishes. Based on this idea, we construct a theory of graph complexes
and cocycles, so that higher-loop invariants can be defined by two different
but equivalent methods: the graph cocycles without self-loops as in
Bott--Cattaneo's theory, and the ones with self-loops that extend
Cattaneo--Shimizu's 2-loop invariants. As a consequence, we prove that the
generating series of Chern--Simons perturbation theory gives rise to
topological invariants for framed 3-manifolds in our setting, which admits a
formula in terms of only trivalent graphs without self-loops.
