On Cayley Graphs of Skew Braces and Their Distances

Source abstract

We introduce the left and right Cayley graphs of a finite skew brace $(A,\cdot, \circ)$ with a connection set $T\subset A$, as a directed multigraph with additive and multiplicative edges, depicting the operations $\cdot$ and $\circ$ respectively. We compute the distance from $a$ to $a\circ b$ in these Cayley graphs in terms of factorization lengths over $T$. As an application, we provide a characterization of the ideals of a skew brace in terms of their corresponding Cayley graphs.