2023/10/18 by Ramesh Prasad Panda, Panda, Ramesh Prasad
Engineering · Mathematics · #graph theory and CDMA systems #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2310.11809
Let Γ be an undirected and simple graph. A set S of vertices in Γ is called a cyclic vertex cutset of Γ if Γ- S is disconnected and has at least two components containing cycles. If Γ has a cyclic vertex cutset, then it is said to be cyclically separable. The cyclic vertex connectivity of Γ is the minimum of cardinalities of the cyclic vertex cutsets of Γ. The power graph P(G) of a group G is the undirected and simple graph whose vertices are the elements G and two vertices are adjacent if one of them is the power of other in G. In this paper, we first characterize the finite p -groups (p is a prime number) whose power graphs are cyclically separable in terms of their maximal cyclic subgroups. Then we characterize the finite p -groups whose power graphs have equal vertex connectivity and cyclic vertex connectivity.