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.Let G be a finite, connected graph with n vertices, and let A be the adjacency matrix of G. An arithmetical structure on G is given by a pair of vectors (d, r) ∈ (Z >0 ) n × (Z >0 ) n for which the

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“…We have already seen what happens with paths and cycles. Critical groups associated to arithmetical structures on bident graphs D n are analyzed in [4,Section 5]. In particular, the authors show that for any r, the matrix L(G, r) has an (n − 2) × (n − 2) minor equal to 1 and use Corollary 1 to show that K (G; r) is cyclic.…”

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