I'm studying chip-firing processes on graphs and a central objet is the jacobian group of the underlying graph, the torsion component of the Picard group of the graph. We know for example that $Jac(C\_n) = \mathbb Z\_n$, where $C\_n$ is the cycle graph on $n$ vertices. Doing some computations for $C\_n$ I found for example that

$Jac(C\_n \bigoplus\_1 C\_m) = \mathbb Z\_{n\cdot m-1}$ if $C\_n \bigoplus\_1 C\_m$ is the graph obtained by glueing$C\_n$ and $C\_m$ on $1$ edge.

$\#Jac \left(C\_n \bigoplus\_k C\_m\right) = n\cdot m - k^2$ if $C\_n \bigoplus\_k C\_m$ is the graph obtained by glueing $C\_n$ and $C\_m$ on $k$ consecutive and different edges

$\#Jac \left(C\_n \bigoplus\_k C\_m \bigoplus\_k C\_r\right) = (n\cdot m \cdot r -k^2)\cdot (m+n+r) - 2\cdot k^3$ if $C\_n \bigoplus\_k C\_m \bigoplus\_k C\_r $ is the graph obtained by glueing $C\_n$ and $C\_m$ on $k$ consecutive and different edges and then gluing $C\_r$ on $k$ consecutive and different edges

What kind of general results do we know in that direction?

I would be glad to have some info on that and maybe some pointers to papers.

Thanks