Engel and co-Engel graphs of finite groups

Let $G$ be a group. Associate a directed graph (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 by ignoring directions; the co-Engel graph 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 (unlike what happens for the power graph) 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. The isolated vertices of the co-Engel graph form the set $L(G)$ be the set of all left Engel elements of $G$. In a finite group $G$, $L(G)$ is the Fitting subgroup of $G$ (a result of Abdollahi). We realize the induced subgraph of co-Engel graphs of certain finite non-Engel groups $G$ induced by $G \setminus L(G)$. The reduced co-Engel graph is obtained by deleting the isolated vertices. We compute genus, various spectra, energies and Zagreb indices of the reduced co-Engel graphs for those groups. As a consequence, we determine (up to isomorphism) all finite non-Engel group $G$ such that the clique number of the co-Engel graph is at most $4$ and the graph is toroidal or projective. Further, we show that the graph is super integral and satisfies the E-LE conjecture and the Hansen–Vuki{č}evi{ć} conjecture for the groups considered in this paper.

💡 Research Summary

The paper investigates three closely related graphs that can be built from a finite group (G): the Engel digraph (\overrightarrow{E}(G)), its undirected version (E(G)), and the complement of the latter, called the co‑Engel graph (E_c(G)). In the Engel digraph an ordered pair ((x,y)) is joined by an arc if there exists a positive integer (k) such that the iterated commutator (