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Counting Arithmetical Structures on Paths and Cycles
- Source :
- Discrete Math. 341 (2018), no. 10, 2949-2963
- Publication Year :
- 2017
-
Abstract
- Let $G$ be a finite, simple, connected graph. An arithmetical structure on $G$ is a pair of positive integer vectors $\mathbf{d},\mathbf{r}$ such that $(\mathrm{diag}(\mathbf{d})-A)\mathbf{r}=0$, 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 cokernels of the matrices $(\mathrm{diag}(\mathbf{d})-A)$). 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 $\binom{2n-1}{n-1}$, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.
- Subjects :
- Mathematics - Combinatorics
Mathematics - Number Theory
Subjects
Details
- Database :
- arXiv
- Journal :
- Discrete Math. 341 (2018), no. 10, 2949-2963
- Publication Type :
- Report
- Accession number :
- edsarx.1701.06377
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.disc.2018.07.002