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 $G\backslash L(G)$ as vertices of $\mathcal{E}_G$ and join two distinct vertices $x$ and $y$ whenever $[x,_k y]\not=1$ and $[y,_k x]\not=1$ for all positive integers $k$. We call $\mathcal{E}_G$, the Engel graph of $G$. In this paper we study the graph theoretical properties of $\mathcal{E}_G$.