Counting arithmetical structures on paths and cycles

Benjamin Braun, Hugo Corrales, Scott Corry, Luis David García Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, Carlos E. Valencia

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

© 2018 Elsevier B.V. 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 [Formula presented], and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.
Original languageAmerican English
Pages (from-to)2949-2963
Number of pages2652
JournalDiscrete Mathematics
DOIs
StatePublished - 1 Oct 2018
Externally publishedYes

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