Conférencier: Joel Louwsma (Niagara University)

An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Associated to each arithmetical structure is a finite abelian group known as its critical group, which may be regarded as a generalization of the sandpile group of a graph. We present results about the groups that occur as critical groups of arithmetical structures on star graphs, complete graphs, and trees. We also present results about how critical groups are transformed under a generalized star-clique operation. These results are joint work with subsets of Kassie Archer, Abigail Bishop, Alexander Diaz-Lopez, Luis Garcia Puente, and Darren Glass.