Ivica Bošnjak – Some New Results Concerning Graphs Associated to Groups

University of Novi Sad, Faculty of Sciences, Department of Mathematics and Informatics

The power graph of a group G is a graph with the vertex set G where vertices x and y are joined if one is a power of the other. The enhanced power graph is a graph with the same vertex set, in which two vertices are joined if they belong to a same cyclic subgroup. We show that two groups with isomorphic power graphs have isomorphic enhanced power graphs too. We give a sufficient condition for a finite group to have perfect enhanced power graph. We determine all symmetric and all alternative groups with perfect enhanced power graphs.
This is a joint work with Rozalia Madarasz and Samir Zahirović.