Spectral Radii of Arithmetical Structures on Cycle Graphs

TL;DR

This paper investigates the spectral radii of arithmetical structures on cycle graphs, identifying those that maximize and minimize the spectral radius among all such structures.

Contribution

It determines the extremal arithmetical structures on cycle graphs with respect to the spectral radius, a novel characterization in this context.

Findings

Identified arithmetical structures with maximum spectral radius.

Identified arithmetical structures with minimum spectral radius.

Provided explicit formulas or characterizations for these extremal structures.

Abstract

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 $(\text{diag}(\mathbf{d})-A_G)\cdot \mathbf{r}=\textbf{0},$ 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 $\mathcal{C}_n.$