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Arithmetical structures on bidents | | Kassie Archer
; Abigail Bishop
; Alexander Diaz-Lopez
; Luis David Garcia Puente
; Darren Glass
; Joel Louwsma
; | | Date: |
4 Mar 2019 | | Abstract: | An arithmetical structure on a finite, connected graph $G$ is a pair of
vectors $(mathbf{d}, mathbf{r})$ with positive integer entries for which
$(operatorname{diag}(mathbf{d}) - A)mathbf{r} = mathbf{0}$, where $A$ is
the adjacency matrix of $G$ and where the entries of $mathbf{r}$ have no
common factor. The critical group of an arithmetical structure is the torsion
part of the cokernel of $(operatorname{diag}(mathbf{d}) - A)$. 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. | | Source: | arXiv, 1903.1393 | | Services: | Forum | Review | PDF | Favorites | |
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