Power Graphs of Cyclic and Dihedral Groups: Structure and Extremal Parameters

Abstract

The (undirected) power graph $\mathcal{P}(G)$ of a finite group $G$ has vertex set $G$ and an edge $\{x,y\}$ whenever one of $x,y$ is a positive power of the other. Power graphs were introduced in directed form by Kelarev--Quinn and, in the undirected group setting, by Chakrabarty--Ghosh--Sen, and have since been studied widely .
In this paper we give an explicit, computation-friendly treatment of $\mathcal{P}(C_n)$ and $\mathcal{P}(D_{2n})$ from a common viewpoint.
For $C_n$ we show that adjacency is governed by divisibility of element orders, identify $\mathcal{P}(C_n)$ as a blow-up of the comparability graph of the divisor lattice, and derive closed formulas for degrees and edge counts. We also obtain exact expressions for the clique and chromatic numbers as the maximum totient-weight of a divisor chain, and for the independence number as the width of the divisor poset.
For $D_{2n}$ we prove a sharp decomposition: $\mathcal{P}(D_{2n})$ is obtained from $\mathcal{P}(C_n)$ by attaching $n$ pendant leaves at the identity. This yields immediate transfer principles for many invariants and, in particular, an exact independence formula
$\alpha(\mathcal{P}(D_{2n}))=n+W'(n)$, where $W'(n)$ is the width of the divisor poset of $n$ with $1$ removed.
We conclude with algorithmic remarks showing how the main parameters can be computed efficiently from the prime factorization of $n$.