Arithmetical structures on bidents

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. (C) 2020 Elsevier B.V. All rights reserved.