Spectral Radii of Arithmetical Structures on Cycle Graphs

Let $G$ be a finite, connected graph. An arithmetical structure on $G$ is a pair of positive integer-valued vectors $(\mathbf{d},\mathbf{r})$ such that $(\text{diag}(\mathbf{d})-A_G)\cdot \mathbf{r}=\mathbf{\text{0}},$ where the entries of $\mathbf{r}$ have $\gcd$ 1 and $A_G$ is the adjacency matrix of $G$. In this article we find the arithmetical structures that maximize and minimize the spectral radius of $(\text{diag}(\mathbf{d})-A_G)$ among all arithmetical structures on the cycle graph ${\mathcal{C}}_n\mathrm{.}$