  An arithmetical structure on a finite, connected graph without loops is an
assignment of positive integers to the vertices that satisfies certain
conditions. Associated to each of these is a finite abelian group known as its
critical group. We show how to determine the critical group of an arithmetical
structure on a star graph or complete graph in terms of the entries of the
arithmetical structure. We use this to investigate which finite abelian groups
can occur as critical groups of arithmetical structures on these graphs.
