On the difference graph of power graphs of finite groups

Abstract

The power graph of a finite group G is the simple undirected graph with vertex set G whose two vertices are adjacent if one is a power of  the other. The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is the group G whose two  vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we investigate the difference graph D(G) of a finite group G, which is the difference of the enhanced power graph and the power graph of G with all isolated vertices  removed. We first characterize an arbitrary finite group G such that D(G) is a chordal graph, star graph, dominatable, threshold graph,  and split graph. From this, we conclude that the latter four graph classes are equal for D(G). By applying these results, we classify the  nilpotent groups G such that D(G) belong to the aforementioned five graph classes. This shows that all these graph classes are equal for  D(G) when G is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of n such that the difference  graphs of the symmetric group Sn and alternating group An are cograph, chordal, split, and threshold.