Graphs defined on groups

TL;DR

This paper explores various automorphism-invariant graphs on groups, analyzing their structure, subgraphs, automorphisms, and reductions, with a focus on twin vertices, cographs, and the influence of the Gruenberg--Kegel graph.

Contribution

It provides a comprehensive analysis of multiple graphs on groups, their automorphism properties, twin reductions, and the role of the Gruenberg--Kegel graph, offering new insights into their structure and symmetries.

Findings

Twin vertices form equivalence classes affecting automorphism groups.

Twin reduction leads to a unique simplified graph, the cograph.

The Gruenberg--Kegel graph influences other group-based graphs.

Abstract

These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are the power graph, enhanced power graph, deep commuting graph, commuting graph, and non-generating graph, though I give a briefer discussion of the nilpotence and solvability graphs, and make some remarks on more general graphs. Aspects to be discussed include induced subgraphs, forbidden subgraphs, connectedness, and automorphism groups. We can also ask about the graphs formed by the edges in one graph but not in an earlier graph in the hierarchy. I have included some results on intersection graphs of subgroups of various types, which are often in a dual relation to one of the other graphs considered. Another actor is the Gruenberg--Kegel graph, or prime graph, of a group: this very small graph influences various graphs defined on the group. I say little about Cayley graphs, since (except in special cases) these are not invariant under the automorphism group of $G$. The graphs all have the property that they contain \emph{twins}, pairs of vertices with the same neighbours (save possibly one another). Being equal or twins is an equivalence relation, and the automorphism group of the graph has a normal subgroup inducing the symmetric group on each twin class. For some purposes, we can merge twin vertices and get a smaller graph. Continuing until no further twins occur, the result is independent of the reduction, and is the $1$-vertex graph if and only if the original graph is a \emph{cograph}. So I devote a section to cographs and twin reduction, and another to consequences for automorphism groups. There are briefer discussions of related matters.