Engel graph associated with a group

Abstract

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,k y] ≠ 1 and [y,k x] ≠ 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. © 2007 Elsevier Inc. All rights reserved.