Some new results concerning power graphs and enhanced power graphs of groups (Q6611010)

The directed power graph of a group \(\mathcal{G}\) is the simple directed graph whose vertex set is \(G\), and in which \(x \to y\) if \(y\) is a power of \(x\). \NIts underlying simple graph is called the power graph of the group. The directed power graph was introduced by \textit{A. V. Kelarev} and \textit{S. J. Quinn} [J. Algebra 251, No. 1, 16--26 (2002; Zbl 1005.20043)]. \NThe power graph of \(\mathcal{G}\), denoted by \(\mathcal{P}(\mathcal{G})\), is the underlying simple graph. The enhanced power graph \(\mathcal{P}_e ( \mathcal{G})\) of \(\mathcal{G}\) is the simple graph with vertex set \(G\) in which two elements are adjacent if they generate a cyclic subgroup.\NThe authors in this paper show that if two groups have isomorphic power graphs, then they also have isomorphic enhanced power graphs. They also answer negatively the question concerning whether all finite groups possess a perfect enhanced power graph when its order is divisible by at most two primes. However, they show that for any \(n \geq 0\) and prime numbers \(p\) and \(q\), every group of order \(p^n q\) and \(p^2 q^2\) has perfect enhanced power graph and obtain a complete characterization of symmetric and alternative groups with perfect enhanced graphs. They conclude the article with a probe on a difference graph which is a difference between the enhanced graph and power graph and prove that groups whose difference graphs are perfect have perfect enhanced graphs.