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An arithmetical structure on a finite, connected graph without loops is givenby an assignment of positive integers to the vertices such that, at eachvertex, the integer there is a divisor of the sum of the integers at adjacentvertices, counted with multiplicity if the graph is not simple. Associated toeach arithmetical structure is a finite abelian group known as its criticalgroup. Keyes and Reiter gave an operation that takes in an arithmeticalstructure on a finite, connected graph without loops and produces anarithmetical structure on a graph with one fewer vertex. We study how thisoperation transforms critical groups. We bound the order and the invariantfactors of the resulting critical group in terms of the original arithmeticalstructure and critical group. When the original graph is simple, we determinethe resulting critical group exactly.

Critical groups of arithmetical structures under a generalized star-clique operation