Graph jacobian group

Question

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

Answer 1

I myself and a former professor of mine actually did quite a bit of work on this subject, and I continue to work on it as well. We wrote a paper on Graph Jacobians and what happens under certain 'gluing operations'. The phenomena you've described above was something I found as well.

Here's the link to our paper if you're interested: https://arxiv.org/abs/2008.03761

This is a wider subject than you might think, and our paper does reference other papers if you're interested in reading those, so I highly recommend you check it out!

Answer 2

I apologize for not elaborating further on my answer!

The gluing operations we studied mostly focused on identifying common subgraphs between two graphs. A popular choice for us were paths or simply common edges. We note the distinction between connecting vertices rather than edges of common subgraphs and how the results change quite drastically, suggesting the story of the Jacobian lies in the edges (duh).

Other "operations" in the paper include:

1. Taking a cycle graph and simply turning every edge into two parallel edges
2. Fan Graphs which are a specific case of our Prop. 4.6 which is an inductive process on computing Jacobians. The graphs are cycle graphs linked together in a chain where each cycle is glued to the previous cycle along 1 edge.
3. Connecting two distinct pairs of vertices on a cycle graph
4. Inscribing a 'triangle' within a cycle graph

Clearly we focused on Jacobians involving cycle graphs. Our main theorems leveraged planar graphs and how their duals have isomorphic Jacobians, which is pretty remarkable. This lead us to Prop. 4.4 where we defined a "Cycle Matrix" which can also compute the Jacobian.

My own direction is to continue studying the role of cycles in a graph and how they are the "important chunks" of the graph with regards to the Jacobian. It is heavily linked to homology and sets the algebra necessary for a lot of computation. Similarly remarkable is the fact that the Laplacian is not the only tool for computing Jacobians; one can use fundamental cuts/cycles to do the same. Depending on the graph, one might be more/less effective and efficient.

In essence, the Jacobian is a way of measuring the "algebra of wave forms" across the network. In the case of chip firing and divisors, which was our original foundation as well, you can think of the process of chip firing as some kind of wave like object that propagates through the graph. The chips could be thought of as "energy" and the edges are a way of displacing that energy. That is, any one divisor is like a "moment" or snapshot in the propagation of the wave, and the equivalence class in the Jacobian group ( or equivalently Picard Group) is like the entire "motion" of that wave. The classes encapsulate the behavior of the wave through the graph, and each representative is like a different state of the wave. The Jacobian then records how we can add these wave forms together to get new waves, and it detects that algebra of wave addition. This is my personal viewpoint on looking at these things.

For example, trees have trivial Jacobians because you can think about any one wave form propagating through the tree, and once it hits a leaf it just "cancels" itself. Every unique path gives the wave no choice but to die at the end of it. However, introducing a cycle in the network gives the wave freedom to "slosh around" and move more freely. This is why cycles tell us everything we need to know.

I hope this was a far more fruitful answer than my original response.