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Counting arithmetical structures on paths and cycles
Braun, Benjamin ; Corrales, Hugo ; Corry, Scott ; Puente, Luis David GarcÃa ; Glass, Darren ; Kaplan, Nathan ; Martin, Jeremy L. ; Musiker, Gregg ; Valencia, Carlos E.
Braun, Benjamin
Corrales, Hugo
Corry, Scott
Puente, Luis David GarcÃa
Glass, Darren
Kaplan, Nathan
Martin, Jeremy L.
Musiker, Gregg
Valencia, Carlos E.
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Abstract
Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag(d)-A)r = 0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag(d)-A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients C(2n-1,n-1), and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.
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Date
2018-07-27
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Elsevier
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Keywords
Arithmetical graph, Ballot number, Catalan number, Critical group, Sandpile group, Laplacian
Citation
Benjamin Braun, Hugo Corrales, Scott Corry, Luis David GarcÃa Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, Carlos E. Valencia, "Counting arithmetical structures on paths and cycles", Discrete Mathematics, Volume 341, Issue 10, 2018, Pages 2949-2963, ISSN 0012-365X, https://doi.org/10.1016/j.disc.2018.07.002.