On the Diameter of Undirected Cayley Graphs of Finite Abelian Groups

TL;DR

This paper investigates the diameter of undirected Cayley graphs of finite abelian groups, providing a complete characterization for cyclic groups and partial results for noncyclic groups, advancing understanding of their structural properties.

Contribution

It offers a complete classification for cyclic groups and proposes a conjecture with partial results for noncyclic groups regarding Cayley graph diameters.

Findings

Complete characterization for cyclic groups

Partial results and conjecture for noncyclic groups

Advances understanding of Cayley graph diameters in abelian groups

Abstract

Let $s$ be a positive integer. Our goal is to find all finite abelian groups $G$ that contain a $2$-subset $A$ for which the undirected Cayley graph $\Gamma(G,A)$ has diameter at most $s$. We provide a complete answer when $G$ is cyclic, and a conjecture and some partial answers when $G$ is noncyclic.