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arXiv · 2208.00743

On the power graph of a certain gyrogroup

Yogendra Singh·Anand Kumar Tiwari·Fawad Ali·Mani Shankar Pandey
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Abstract

The power graph $P(G)$ of a group $G$ is a simple graph with the vertex set $G$ such that two distinct vertices $u,v \in G$ are adjacent in $P(G)$ if and only if $u^m = v$ or $v^m = u$, for some $m \in \mathbb{N}$. The purpose of this paper is to introduce the notion of a power graph for gyrogroups. Using this, we investigate the combinatorial properties of a certain gyrogroup, say $G(n)$, of order $2^n$ for $n \geq 3$. In particular, we determine the Hamiltonicity and planarity of the power graph of $G(n)$. Consequently, we calculate distant properties, resolving polynomial, Hosoya and reciprocal Hosoya polynomials, characteristic polynomials, and the spectral radius of the power graph of $G(n)$.

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Yogendra Singh, Anand Kumar Tiwari, Fawad Ali, Mani Shankar Pandey. 2022-08-01. On the power graph of a certain gyrogroup. https://arxiv.org/abs/2208.00743

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