dc.contributor.author | Braun, Benjamin | |
dc.contributor.author | Corrales, Hugo | |
dc.contributor.author | Corry, Scott | |
dc.contributor.author | Puente, Luis David García | |
dc.contributor.author | Glass, Darren | |
dc.contributor.author | Kaplan, Nathan | |
dc.contributor.author | Martin, Jeremy L. | |
dc.contributor.author | Musiker, Gregg | |
dc.contributor.author | Valencia, Carlos E. | |
dc.date.accessioned | 2021-02-15T22:42:52Z | |
dc.date.available | 2021-02-15T22:42:52Z | |
dc.date.issued | 2018-07-27 | |
dc.identifier.citation | Benjamin Braun, Hugo Corrales, Scott Corry, Luis David García 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. | en_US |
dc.identifier.uri | http://hdl.handle.net/1808/31429 | |
dc.description.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. | |
dc.publisher | Elsevier | en_US |
dc.rights | © 2018 Elsevier B.V. All rights reserved. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. | en_US |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | en_US |
dc.subject | Arithmetical graph | en_US |
dc.subject | Ballot number | en_US |
dc.subject | Catalan number | en_US |
dc.subject | Critical group | en_US |
dc.subject | Sandpile group | en_US |
dc.subject | Laplacian | en_US |
dc.title | Counting arithmetical structures on paths and cycles | en_US |
dc.type | Article | en_US |
kusw.kuauthor | Martin, Jeremy L. | |
kusw.kudepartment | Mathematics | en_US |
dc.identifier.doi | 10.1016/j.disc.2018.07.002 | en_US |
kusw.oaversion | Scholarly/refereed, author accepted manuscript | en_US |
kusw.oapolicy | This item meets KU Open Access policy criteria. | en_US |
dc.rights.accessrights | openAccess | en_US |