Characterizations of $p$-groups whose power graphs satisfy certain connectivity conditions

Abstract

Let $\Gamma$ be an undirected and simple graph. A set $S$ of vertices in $\Gamma$ is called a {cyclic vertex cutset} of $\Gamma$ if $\Gamma -S$ is disconnected and has at least two components containing cycles. If $\Gamma$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. The {cyclic vertex connectivity} of $\Gamma$ is the minimum of cardinalities of the cyclic vertex cutsets of $\Gamma$. The {power graph} $\mathcal{P}(G)$ 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.