In Summer 2019 David Duncan and I were co-mentors for one of the NSF REU projects at JMU (NSF-DMS 1560151).

From left to right in the photo above we have David Duncan, Jawahar Madan, Angela Vichitbandha, Christina Shatford, Eric Piato, Wesley Engelbrecht, Josh Ducey.

These talented and hardworking students had an extremely productive 8 weeks. I certainly learned a lot! Some of the topics they investigated:

• What can be deduced about the critical group of a strongly regular graph from its parameters alone? Can this information be used to show that a graph with a particular parameter set cannot exist?
• How can the critical group be used to distinguish vertices in a graph? Delete a vertex and its neighbors and compute the critical group of the resulting graph. Does the group change as the deleted vertex varies?
• Can we find any results connecting the critical group, or generally the Smith normal form of L+cI, to that of a subgraph? Can any results of this sort hold as the graph becomes more symmetric?
• There are nice pictures one finds in the literature of particular elements of the sandpile group of square lattice graphs (for example, the stable and recurrent configuration representing the identity element). Is there a similar limiting behavior when we consider a hexagonal lattice, or other types of graphs that can be scaled similarly?
• It is known that a planar graph and its dual have isomorphic critical groups. What can be said when we embed a graph on a torus or other surface? What conditions on the embedding lead to something interesting?
• Given a particular sandpile configuration on a graph, we know we can stabilize it through chip-firing. In Levine and Propp's article the odometer (firing sequence function) is characterized as the unique minimal integral solution to a system of inequalities. Allowing real-valued functions as solutions to the system, there is a unique minumum no larger than the odometer. When these two solutions differ, the sandpile is divisible. Can the divisible sandpiles be shown to exist on a particular graph? Can they be characterized? Extremely interesting are the PDE techniques that can be brought to bear here.

Do yourself a favor and have a look at their excellent report.
Some of the SRG results were collected into a paper.
The students gave a very nice presentation to the JMU Math and Stat Department. Some have given talks at their home universities or will be presenting at JMM 2020.

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