Super graphs on groups, I

TL;DR

This paper explores a hierarchy of graphs on finite groups, including power, enhanced power, and commuting graphs, modified by equivalence relations, to classify groups and analyze graph properties like completeness and dominant vertices.

Contribution

It introduces a second dimension to the graph hierarchy via equivalence relations, characterizes when these graphs are complete, and identifies conditions for group classes based on graph properties.

Findings

Characterized groups with complete graphs

Identified conditions for dominant vertices

Analyzed universality, perfectness, and clique number

Abstract

Let $G$ be a finite group. A number of graphs with the vertex set $G$ have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful to study them together. In addition, several authors have considered modifying the definition of these graphs by choosing a natural equivalence relation on the group such as equality, conjugacy, or equal orders, and joining two elements if there are elements in their equivalence class that are adjacent in the original graph. In this way, we enlarge the hierarchy into a second dimension. Using the three graph types and three equivalence relations mentioned gives nine graphs, of which in general only two coincide; we find conditions on the group for some other pairs to be equal. These often define interesting classes of groups, such as EPPO groups, $2$-Engel groups, and Dedekind groups. We study some properties of graphs in this new hierarchy. In particular, we characterize the groups for which the graphs are complete, and in most cases, we characterize the dominant vertices (those joined to all others). Also, we give some results about universality, perfectness, and clique number. The paper ends with some open problems and suggestions for further work.