Some New Results Concerning Power Graphs and Enhanced Power Graphs of Groups

TL;DR

This paper explores the relationships between power graphs and enhanced power graphs of groups, proving isomorphism implications, examining perfection properties across various group orders, and characterizing specific groups with perfect enhanced graphs.

Contribution

It establishes that isomorphic power graphs imply isomorphic enhanced power graphs, and characterizes groups with perfect enhanced power graphs, including symmetric and alternating groups.

Findings

Isomorphic power graphs imply isomorphic enhanced power graphs.

Finite groups of order p^n q and p^2 q^2 have perfect enhanced power graphs.

Symmetric and alternating groups are characterized by their perfect enhanced power graphs.

Abstract

The directed power graph $\vec{\mathcal P}(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ such that $x\rightarrow y$ if $y$ is a power of $x$. The power graph of $\mathbf G$, denoted by $\mathcal P(\mathbf G)$, is the underlying simple graph. The enhanced power graph $\mathcal P_e(\mathbf G)$ of $\mathbf G$ is the simple graph with vertex set $G$ in which two elements are adjacent if they generate a cyclic subgroup. In this paper, it is proven that, if two groups have isomorphic power graphs, then they have isomorphic enhanced power graphs, too. It is known that any finite nilpotent group of order divisible by at most two primes has perfect enhanced power graph. We investigated whether the same holds for all finite groups, and we have obtained a negative answer to that question. Further, we proved that, for any $n\geq 0$ and prime numbers $p$ and $q$, every group of order $p^nq$ and $p^2q^2$ has perfect enhanced power graph. We also give a complete characterization of symmetric and alternative groups with perfect enhanced graphs.