Arithmetical structures on bidents

Abstract An arithmetical structure on a finite, connected graph G is a pair of vectors ( d , r ) with positive integer entries for which ( diag ( d ) − A ) r = 0 , where A is the adjacency matrix of G and where the entries of r have no common factor. The critical group of an arithmetical structure is the torsion part of the cokernel of ( diag ( d ) − A ) . In this paper, we study arithmetical structures and their critical groups on bidents, which are graphs consisting of a path with two “prongs” at one end. We give a process for determining the number of arithmetical structures on the bident with n vertices and show that this number grows at the same rate as the Catalan numbers as n increases. We also completely characterize the groups that occur as critical groups of arithmetical structures on bidents.