G-graph and G-gyro-graph

Abstract

In this short note, we survey several results concerning the structure of certain graphs, introduced by the fourth-named author, that can be associated to groups and gyro-groups, which are algebraic structures satisfying a twisted form of associativity. Furthermore, we include an original result, showing that the so-called G-gyro-graph associated to a gyro-group G endowed with a subset S ⊂ G is connected if and only if S is a left generating set of G. Moreover some properties and examples of both G-graphs and G-gyro-graphs are presented.