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.
Archer, Kassie, Abigail C. Bishop, Alexander Diaz-Lopez, Luis D. García Puente, Darren Glass, and Joel Louwsma. "Arithmetical Structures on Bidents." Discrete Mathematics 343, no. 7 (2020): 111850.
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