Critical and minimal connectivity of power graphs of finite groups

Ramesh Prasad Panda

Published 2018-05-30Version 1

The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. In this article, we show that $\mathcal{G}(\mathbb{Z}_n)$ is not critically vertex connected when $n$ is a product of two or three distinct primes. We show that $\mathcal{G}(D_n)$ is not critically vertex (edge) connected. We prove that $\mathcal{G}(Q_n)$ is not critically vertex connected and classify $Q_n$ such that $\mathcal{G}(Q_n)$ is critically edge connected. We classify $p$-groups whose power graphs are critically vertex (edge) connected. We obtain a characterization for power graphs of finite groups that are critically edge connected. We supply some necessary criterion for minimally vertex connected graphs. Then we classify finite groups whose power graphs are minimally vertex connected. We classify finite groups of odd order whose power graphs are minimally edge connected. We show that $\mathcal{G}(D_n)$ and $\mathcal{G}(Q_n)$ are not minimally edge connected. Further, we classify abelian $p$-groups whose power graphs are minimally edge connected.