On the power graph of a certain gyrogroup

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 $n\ge 3$. 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 $G(n)$.

15 pages, 4 figures