Counting Arithmetical Structures on Paths and Cycles

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 $\left(\genfrac{}{}{0ex}{}{2n-1}{n-1}\right)$, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.

Jesse Geneson discovered an error in the final sentence of Proposition 33. The sentence should read, "In particular, the number of arithmetical $r$-structures on $C_n$ with $r_i=1$ is the Catalan number $C_{n-1}$." The published version mistakenly refers to $d$-structures instead of $r$-structures