Co-Engel graphs of certain finite non-Engel groups

Let G be a group. Associate a graph E_G (called the co-Engel graph of G) with G whose vertex set is G and two distinct vertices x and y are adjacent if [x, _ky] 6= 1 and [y, _kx] 6= 1 for all positive integer k. This graph, under the name "Engel graph", was introduced by Abdollahi [2].

Let L(G) be the set of all left Engel elements of G. In this paper, we realize the induced subgraph of co-Engel graphs of certain finite non-Engel groups G induced by G\L(G). We write E⁻ (G) to denote the subgraph of E_G induced by G\L(G). We also compute genus, various spectra, energies and Zagreb indices of E⁻ (G) for those groups. As a consequence, we determine (up to isomorphism) all finite non-Engel group G such that the clique number ω(E⁻(G)) is at most 4 and E⁻ (G) is toroidal or projective. Further, we show that E⁻(G) is super integral and satisfies the E-LE conjecture and the Hansen–Vukičević conjecture for the groups considered in this paper.

We also look briefly at the directed Engel graph, with an arc x → y if [y, _kx] = 1 for some k. We show that, if G is a finite soluble group, this graph either is the complete directed graph (which occurs only if G is nilpotent), or has pairs of vertices joined only by single arcs.