Name
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 2n−1n−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. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1016/j.disc.2018.07.002 | Discrete Mathematics |
Keywords | Field | DocType |
Arithmetical graph,Ballot number,Catalan number,Critical group,Sandpile group,Laplacian | Integer,Adjacency matrix,Discrete mathematics,Arithmetic function,Combinatorics,Matrix (mathematics),Catalan number,Enumeration,Binomial coefficient,Connectivity,Mathematics | Journal |
Volume | Issue | ISSN |
341 | 10 | 0012-365X |
Citations | PageRank | References |
0 | 0.34 | 2 |
Authors | ||
9 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Benjamin Braun | 1 | 7 | 3.80 |
| hugo corrales | 2 | 0 | 0.68 |
| Scott Corry | 3 | 0 | 0.68 |
| Luis David García-Puente | 4 | 81 | 10.52 |
| Darren B. Glass | 5 | 0 | 2.70 |
| Nathan Kaplan | 6 | 0 | 0.34 |
| Jeremy L. Martin | 7 | 59 | 10.01 |
| Gregg Musiker | 8 | 11 | 4.86 |
| Carlos E. Valencia | 9 | 11 | 4.99 |