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

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, integer there divisor sum adjacent vertices, counted with multiplicity if not simple. Associated finite abelian group known as its critical group. Keyes and Reiter gave operation that takes in produces one fewer vertex. We study how this transforms groups. bound order invariant factors resulting terms original When simple, we determine exactly.