Let $(V,E)$ be a graph which we view as a $1$-dimensional simplicial complex. The chain group $C_1$ is the free abelian group over $E$ as a set of free generators and the first homology group $H_1$ is (since $C_2$ is trivial) a subgroup of $C_1$ generated by the edge cycles of the graph.
In this particular case, we may thus consider the quotient $C_1/H_1$. I guess this object must have been studied and probably has some name. I would be grateful for any reference.