Graphs, group actions and skew braces

A skew brace is an interlinking of two group structures on the same set, where one group (called the multiplicative group) acts via the so-called lambda-action by automorphisms on the other (called the additive group). These structures play a fundamental role in the combinatorial study of solutions to the Yang--Baxter equation. In this talk, we first define the common divisor graph associated with any action by automorphisms, provide some examples and prove bound for the number of connected components and for the diameter. We then define two common divisor graphs associated with skew braces one related to the lambda-action and the second one that encodes also the conjugation of the additive group. In these cases, we deal with extremal cases. In particular, we classify all skew braces with one of the graphs consisting of two disconnected vertices and with only one vertex.