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Abstract

Let $G$ be a group. The co-Engel graph of $G$) with $G$ whose vertex set is $G$ and two distinct vertices $x$ and $y$ are adjacent if $[x,{}_k y]\neq1$ and $[y,{}_k x] \neq 1$ for all positive integer $k$, where $[x,{}_ky]$ is the iterated commutator $[x,y,y,\ldots,y]$, with $k$ terms $y$ in the expression; usually we delete isolated vertices (these are the left Engel elements). This graph, under the name ``Engel graph'', was introduced by Abdollahi~\cite{aa}. However, we argue that it is more naturally called the ``co-Engel graph''. We compute genus, various spectra, energies and Zagreb indices of co-Engel graphs for groups including the dihedral and generalized quaternion groups and nonabelian groups of order $pq$ where $p$ and $q$ are primes. As a consequence, we determine (up to isomorphism) all finite non-Engel group $G$ such that the clique number of the co-Engel is at most $4$ and the graph is toroidal or projective. Further, we show that the co-Engel graph is super integral and satisfies the E-LE conjecture and the Hansen--Vuki{\v{c}}evi{\'c} conjecture for the groups considered in this paper. We also look briefly at the directed Engel graph, with an arc $x\to y$ 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. We also show that the (directed or undirected) Engel graph of a group $G$ is the lexicographic product of a complete graph of order $Z_\infty(G)$ by the (directed or undirected) Engel graph of $G/Z_\infty(G)$, where $Z_\infty(G)$ is the hypercenter of $G$.