Graphs encoding the generating properties of a finite group

Assume that $G$ is a finite group. For every $a, b inmathbb N,$ we define a graph $Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^acup G^b$ and in which two tuples $(x_1,dots,x_a)$ and $(y_1,dots,y_b)$ are adjacent if and only if $langle x_1,dots,x_a,y_1,dots,y_b rangle =G.$ 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 $G$ are encoded by these graphs.