Graphs encoding the generating properties of a finite group

Assume that G is a finite group. For every a , b ∈ N , we define a graphΓ a , b( G )whose vertices correspond to the elements ofG a ∪ G band in which two tuples ( x 1 , ⋯ , x a ) and ( y 1 , ⋯ , y b ) are adjacent if and only if ⟨ x 1 , ⋯ , x a , y 1 , ⋯ , y b ⟩ = G . 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.