
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 and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of ). 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…
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Taxonomy
TopicsFinite Group Theory Research · graph theory and CDMA systems · Advanced Graph Theory Research
