Enhanced power graph from the power graph of a group

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 $G$.