One can define cochain complexes of (combinatorial) graphs, where each term is a vector space of linear combinations of certain (isomorphism classes of) graphs, and where the differential $d$ is a signed sum of edge contractions. One can understand the kernel of $d$ by understanding the image (and hence the cokernel) of its adjoint $d^*$. It is easy to see that $d^*$ takes a graph to some linear combination of all possible "edge expansions" (some of which may be isomorphic graphs!). The exact values of the coefficients in this linear combination are perhaps less obvious. If one identifies the dual to the graph complex as a vector space of graphs via the pairing $\langle \Gamma_i, \Gamma_j\rangle = \delta_{ij}|\mathrm{Aut}(\Gamma_i)|$, those coefficients turn out to be all $(\pm)1$. (This allows one to deduce the STU and IHX relations in finite-type invariants via graph cohomology.)

In the case of "unitrivalent" graphs (which is all I'm interested in), I have written a somewhat complicated proof justifying these coefficients (or automorphism factors, depending on how you look at it). But if someone else has already written down this proof (or a shorter one, or a more general one), I'd be happy to know about it. Similarly, I'd be happy to get a hint as to why this is obvious, if that's the case.