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 (d,r) such that (diag⁡(d)−AG)⋅r=0, where the entries of r have gcd 1 and AG is the adjacency matrix of G. In this article, we find the arithmetical structures that maximize and minimize the spectral radius of (diag⁡(d)−AG) among all arithmetical structures on the cycle graph Cn.