Engel and co-Engel graphs of finite groups

Let $G$ be a group. Associate a directed graph $\vec{E}(G)$ (called the Engel digraph of $G$) with $G$ whose vertex set is $G$, with an arc $(x,y)$ if $[y, {}_k x]=1$ for some positive integer $k$, where $[y,{}_kx]$ is the iterated commutator $[y,x,x,\ldots,x]$, with $k$ terms $x$ in the expression. From this we define the Engel graph $E(G)$ by ignoring directions; the co-Engel graph $E_c(G)$ is its complement. The co-Engel graph, under the name ``Engel graph'', was introduced by Abdollahi. However, the name we use is more natural. We begin with some general results about the Engel digraph and graph, before turning our attention to the co-Engel graph. Among other things, we show that the undirected Engel graph does not determine the directed version up to isomorphism, though counterexamples seem to be fairly rare: there are just two orders less than $100$ for which this happens. We also prove a universality theorem: every finite digraph is an induced sub-digraph of the Engel digraph of a finite group. The isolated vertices of $E_c(G)$ form the Fitting subgroup $F(G)$ of $G$. In this paper, we realize the induced subgraph of co-Engel graphs of certain finite non-Engel groups $G$ induced by $G \setminus F(G)$. We write $E_c^-(G)$ to denote the subgraph of $E_c(G)$ induced by $G \setminus F(G)$. We also compute genus, various spectra, energies and Zagreb indices of $E_c^-(G)$ for those groups. As a consequence, we determine (up to isomorphism) all finite non-Engel group $G$ such that the clique number of $E_c^-(G)$ is at most $4$ and $E_c^-(G)$ is toroidal or projective. Further, we show that $E_c^-(G)$ is ALQ-integral and satisfies the E-LE conjecture and the Hansen-Vuki{\v{c}}evi{\'c} conjecture for the groups considered in this paper.