























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.
From: Jeremy L. Martin [view email]
[v1]
Mon, 23 Jan 2017 13:27:08 UTC (42 KB)
[v2]
Mon, 13 Feb 2017 17:05:38 UTC (26 KB)
[v3]
Sun, 15 Oct 2017 03:52:32 UTC (24 KB)
[v4]
Mon, 23 Jul 2018 23:56:38 UTC (25 KB)
[v5]
Wed, 1 Jul 2026 15:33:26 UTC (25 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理,仅供阅读参考。 原文来自 — 版权归原作者所有。