On a problem about the connectivity of the proper enhanced power graph of a finite group

چکیده مقاله

Let G be a group with identity element e. The proper power graph and proper enhanced power graph of G, are denoted by ΓP∗(G) and ΓEP∗(G), respectively. Also, the prime graph of G is denoted by Γ _GK(G). In an article, Aalipour et al. (Electronic J. Combin.24(3) (2017) 3–16) asked which groups do have the property that ΓP∗(G) is connected? In this paper, we show that if Γ _GK(G) is disconnected, then ΓP∗(G) and ΓEP∗(G) are disconnected. Moreover, we prove that if G is a nilpotent group which is not a p-group, then ΓEP∗(G) is a connected graph. © 2022, Indian Academy of Sciences.