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 ${\mathcal{E}}_G$ as follows: Take $G\backslash L(G)$asverticesof$$\mathcal{E}$_G$andjointwodistinctvertices$x$and$y$whenever$[x,_k y]\not=1$and$[y,_k x]\not=1$forallpositiveintegers$k$. We call ${\mathcal{E}}_G$, the Engel graph of $G$. In this paper we study the graph theoretical properties of ${\mathcal{E}}_G$.