  We study the effect of Feynman integration and diagrammatic differential
operators on the structure of group-like elements in the algebra generated by
coloured vertex-oriented uni-trivalent graphs. We provide applications of our
results to the study of the LMO invariant, a quantum invariant of manifolds. We
also indicate further situations in which our results apply and may prove
useful. The enumerative approach that we adopt has a clarity that has enabled
us to perceive a number of generalizations.
