The complement of enhanced power graph of a finite group [article]

Parveen, Jitender Kumar
2022 arXiv   pre-print
The enhanced power graph 𝒫_E(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 𝒫̅_̅E̅(̅G̅)̅ 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
more » ... (̅G̅)̅ is bipartite. We show that the graph 𝒫̅_̅E̅(̅G̅)̅ is weakly perfect. Further, we study the subgraph 𝒫̅_̅E̅(̅G̅^̅*̅)̅ of 𝒫̅_̅E̅(̅G̅)̅ induced by all the non-isolated vertices of 𝒫̅_̅E̅(̅G̅)̅. We classify all finite groups G such that the graph is 𝒫̅_̅E̅(̅G̅^̅*̅)̅ is unicyclic and pentacyclic. We prove the non-existence of finite groups G such that the graph 𝒫̅_̅E̅(̅G̅^̅*̅)̅ is bicyclic, tricyclic or tetracyclic. Finally, we characterize all finite groups G such that the graph 𝒫̅_̅E̅(̅G̅^̅*̅)̅ is outerplanar, planar, projective-planar and toroidal, respectively.
arXiv:2207.04641v1 fatcat:crhjdl7cebhv3fphqo7jfv5sym