Engel graph associated with a group

Let G be a non-Engel group and let L(G) be the set of all left Engel elements of G. Associate with G a graph EG as follows: Take G\L(G) as vertices of EG and join two distinct vertices x and y whenever [x,ky]≠1 and [y,kx]≠1 for all positive integers k. We call EG, the Engel graph of G. In this paper we study the graph theoretical properties of EG.