On the power graph of a certain gyrogroup

TL;DR

This paper extends the concept of power graphs from groups to gyrogroups, analyzing their combinatorial properties, including Hamiltonicity, planarity, and spectral characteristics, for a specific gyrogroup of order 2^n.

Contribution

It introduces the notion of power graphs for gyrogroups and investigates their properties, including Hamiltonicity, planarity, and spectral features, for a particular class of gyrogroups.

Findings

Power graph of G(n) is Hamiltonian for n ≥ 3.

Power graph of G(n) is planar for n ≥ 3.

Spectral radius and polynomial invariants of the power graph are explicitly computed.

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 $G(n)$.