A characterization via graphs of the soluble groups in which permutability is transitive

There are different ways to associate to a group a certain graph. In this context, it is interesting to ask for the relations between the structure of the group, given in group-theoretical terms, and the structure of the graphs, given in the language of graph theory.
In this paper we recall some properties of the groups in which permutability is a transitive relation and present a new characterisation of the class of soluble groups in which permutability is a transitive relation in graph-theoretical terms.