The Cayley graph of a quandle

TL;DR

This paper explores the structure of Cayley graphs of various quandles, providing explicit descriptions and properties for classes like Alexander and conjugation quandles, with implications for their connectivity and regularity.

Contribution

It offers new structural insights into Cayley graphs of quandles, including explicit descriptions for Alexander and inner automorphism-based quandles, and characterizes their connected components.

Findings

Cayley graphs of Alexander quandles over finite abelian groups have connected components as cosets of a specific subgroup.

The Cayley graphs of generalized Alexander quandles are regular.

Connected components of Cayley graphs correspond to cosets related to the defining automorphism.

Abstract

In this paper, we investigate structural properties of the Cayley graph of a quandle and describe this graph for several important classes of quandles, including conjugation, Takasaki, dihedral, and Alexander quandles. In particular, we prove that for an Alexander quandle $A_t(G)$ over a finite abelian group $G$, the connected components of the Cayley graph correspond to the cosets of the subgroup $\mathrm{im}(\mathrm{id}-t)$. We also show that the Cayley graphs of generalized Alexander quandles are regular. When the defining automorphism is inner, we give an explicit description of the forward orbits and prove that the connected components correspond to cosets of the subgroup generated by commutators with the defining element.