Common divisor graphs for skew braces

TL;DR

This paper introduces and analyzes two new graphs associated with finite skew braces, revealing their structural properties, restrictions, and classifications, including the number of components, diameters, and special cases with one or two vertices.

Contribution

It defines common divisor graphs for skew braces, establishes bounds on their properties, and classifies skew braces based on these graph structures, extending group theoretic concepts.

Findings

Number of connected components is at most two

Diameter of a connected component is at most four

Classified skew braces with a single-vertex graph into four infinite families

Abstract

We introduce two common divisor graphs associated with a finite skew brace, based on its $\lambda$- and $\theta$-orbits. We prove that the number of connected components is at most two and the diameter of a connected component is at most four. Furthermore, we investigate their relationship with isoclinism. Similarly to its group theoretic inspiration, the skew braces with a graph with two disconnected vertices are very restricted and are determined. Finally, we classify all finite skew braces with a graph with one vertex, where four infinite families arise.