Enhanced power graph from the power graph of a group
Amar S. Pote, Ganesh S. Kadu
TL;DR
The paper addresses constructing the enhanced power graph $\mathscr{P}_e(G)$ from the power graph $\mathscr{P}(G)$ without knowledge of the underlying group $G$. It introduces an arithmetic function $N_a$ that counts closed twins in $\mathscr{P}(G)$ and proves its monotonicity on cyclic subgroups, enabling precise criteria (two main theorems) to decide when a non-edge of $\mathscr{P}(G)$ becomes an edge in $\mathscr{P}_e(G)$. These results yield a simple, practical algorithm (Section 5) to recover $\mathscr{P}_e(G)$ from $\mathscr{P}(G)$ and to derive the difference graph $\mathscr{D}(G)$, thus answering Cameron’s question. The method hinges on analyzing the cyclic-subgroup structure via $N_a=|\widetilde{N(a)}|+1$ and its relation to $|\langle a\rangle|$ through $\psi$, enabling edge augmentation based on witnesses $c$ with $a,b\sim c$.
Abstract
The power graph of a group $G$ is a graph with vertex set $G$, where two distinct vertices $a$ and $b$ are adjacent if one of $a$ and $b$ is a power of the other. Similarly, the enhanced power graph of $G$ is a graph with vertex set $G$, where two distinct vertices are adjacent if they belong to the same cyclic subgroup. In this paper we give a simple algorithm to construct the enhanced power graph from the power graph of a group without the knowledge of the underlying group. This answers a question raised by Peter J. Cameron of constructing enhanced power graph of group $G$ from its power graph. We do this by defining an arithmetical function on finite group $G$ that counts the number of closed twins of a given vertex in the power graph of a group. We compute this function and prove many of its properties. One of the main ingredients of our proofs is the monotonicity of this arithmetical function on the poset of all cyclic subgroups of $G$.