Given a finite group G, the Engel graph of G is a directed graph Gamma(G) encoding pairs of elements satisfying some Engel word. Namely, Gamma(G) is the directed graph, where the vertices are the non-hypercentral elements of G and where there is an arc from x to y if and only if [x,(n)y]=1 for some n is an element of N. From previous work, it is known that, except for a few exceptions, Gamma(G) is strongly connected. In this paper, we give an absolute upper bound on the diameter of Gamma(G), when Gamma(G) is strongly connected.
On the diameter of Engel graphs
Lucchini A.;
2025
Abstract
Given a finite group G, the Engel graph of G is a directed graph Gamma(G) encoding pairs of elements satisfying some Engel word. Namely, Gamma(G) is the directed graph, where the vertices are the non-hypercentral elements of G and where there is an arc from x to y if and only if [x,(n)y]=1 for some n is an element of N. From previous work, it is known that, except for a few exceptions, Gamma(G) is strongly connected. In this paper, we give an absolute upper bound on the diameter of Gamma(G), when Gamma(G) is strongly connected.
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