2022/07/11 by Jitender Kumar, Parveen, Kumar, Jitender
Mathematics · Medicine · Materials Science · #Finite Group Theory Research #Synthesis of Organic Compounds #Synthesis and properties of polymers
paper · pdf · doi:10.48550/arxiv.2207.04641
The enhanced power graph PE(G) of a finite group G is the simple undirected graph whose vertex set is G and two distinct vertices x, y are adjacent if x, y ∈ ⟨ z ⟩ for some z ∈ G. In this article, we give an affirmative answer of the question posed by Cameron [6] which states that: Is it true that the complement of the enhanced power graph PE(G) of a non-cyclic group G has only one connected component apart from isolated vertices? We classify all finite groups G such that the graph PE(G) is bipartite. We show that the graph PE(G) is weakly perfect. Further, we study the subgraph PE(G^*) of PE(G) induced by all the non-isolated vertices of PE(G). We classify all finite groups G such that the graph is PE(G^*) is unicyclic and pentacyclic. We prove the non-existence of finite groups G such that the graph PE(G^*) is bicyclic, tricyclic or tetracyclic. Finally, we characterize all finite groups G such that the graph PE(G^*) is outerplanar, planar, projective-planar and toroidal, respectively.