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arXiv · 2108.06088

On the minimum degree of power graphs of finite nilpotent groups

Ramesh Prasad Panda·Kamal Lochan Patra·Binod Kumar Sahoo
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Abstract

The power graph $\mathcal{P}(G)$ of a group $G$ is the simple graph with vertex set $G$ and two vertices are adjacent whenever one of them is a positive power of the other. In this paper, for a finite noncyclic nilpotent group $G$, we study the minimum degree $\delta(\mathcal{P}(G))$ of $\mathcal{P}(G)$. Under some conditions involving the prime divisors of $|G|$ and the Sylow subgroups of $G$, we identify certain vertices associated with the generators of maximal cyclic subgroups of $G$ such that $\delta(\mathcal{P}(G))$ is equal to the degree of one of these vertices. As an application, we obtain $\delta(\mathcal{P}(G))$ for some classes of finite noncyclic abelian groups $G$.

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Ramesh Prasad Panda, Kamal Lochan Patra, Binod Kumar Sahoo. 2021-08-13. On the minimum degree of power graphs of finite nilpotent groups. https://arxiv.org/abs/2108.06088

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