标题： 关于有限群的幂图满足某些连通性条件的研究 显示英文标题

标题： On finite groups whose power graphs satisfy certain connectivity conditions

摘要： 考虑一个图 $\Gamma$。 如果 $\Gamma - S$ 是不连通的，并且至少有两个包含循环的连通分支，则 $\Gamma$ 中顶点的一个集合 $ S $ 被称为 {循环顶点割集} 的 $\Gamma$。 若$\Gamma$具有循环顶点割集，则称其为{循环可分离的}。 {循环顶点连通性}是$\Gamma$的循环顶点割集的最小基数。 群$G$的幂图$\mathcal{P}(G)$是以顶点集$G$为顶点集的无向简单图，并且当且仅当两个不同顶点中的一个为另一个的正幂时它们相邻。 如果 $G$ 是循环群、二面体群或二循环群，我们确定 $G$ 的阶使得 $\mathcal{P}(G)$ 是循环可分的。然后我们通过 $G$ 的阶来刻画 $\mathcal{P}(G)$ 的顶点连通性和循环顶点连通性的相等性。 显示英文摘要

摘要： Consider a graph $\Gamma$. 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} is the minimum cardinality of a cyclic vertex cutset of $\Gamma$. The power graph $\mathcal{P}(G)$ of a group $G$ is the undirected simple graph with vertex set $G$ and two distinct vertices are adjacent if one of them is a positive power of the other. If $G$ is a cyclic, dihedral, or dicyclic group, we determine the order of $G$ such that $\mathcal{P}(G)$ is cyclically separable. Then we characterize the equality of vertex connectivity and cyclic vertex connectivity of $\mathcal{P}(G)$ in terms of the order of $G$.