Graphs encoding the generating properties of a finite group

Assume that G is a finite group. For every (Formula presented.), we define a graph (Formula presented.) whose vertices correspond to the elements of (Formula presented.) and in which two tuples (Formula presented.) and (Formula presented.) are adjacent if and only if (Formula presented.). We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about G is encoded by these graphs.