Let $G$ be a graph, and define $C_n$ to be the free abelian group on labeled $K_k$ minors of $G$. We can define a boundary map $\delta_n$ from $C_n$ onto $C_{n - 1}$ by taking a $K_k$ minor and mapping it to the sum of its $K_{k - 1}$ minors times $(-1)^i$ where $i$ is the label of the structure removed. I believe that according to this definition, $\delta_{n - 1} \circ \delta_n = 0$, which states this is a chain complex. Can anyone provide citations for previous research into this homology? I tried to find references to this online but, mostly found only discussions of singular and simplicial homology for graphs as CW-complexes.
This particular homology appears naturally as the first "graph-theoretic" homology rather than a topological homology. In particular, I want to know if it plays nicely with graph homomorphism. If I have two chains for graphs $H, G$ with a homomorphism $\varphi$ between them, will this define an induced $\varphi^*$ between the chains. If not, maybe it would be better to look at the family of homomorphism over minors of $K_k$ into $G$, in which case we can apply the fact that the composition of two homomorphism is a homomorphism. But in this case, it would be much harder to calculate the homology groups. Can we further say that these homologies are isomorphic, in a similar way to singular and simplicial homologies are isomorphic?