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This is a Quantitative Study study.

September 23, 2025Open Access

On prime coprime graphs of certain finite groups

Key Points

The prime coprime graph Θ(G) shows Hamiltonicity properties for specific finite groups.
The analysis establishes that Θ(G) has distinct clique numbers for cyclic and dihedral groups.
Graph properties like vertex degree reveal significant patterns within dicyclic groups.
A (k,1)-partition of Θ(G) characterizes the structure for specified orders of the groups.

Abstract

The prime coprime graph Θ (G) of a finite group G is the graph whose vertex set is G and any two distinct vertices are adjacent if the greatest common divisor of their orders is either 1 or a prime. In this paper, we investigate Hamiltonicity, clique number, and vertex degree of Θ (G) for cyclic, dihedral, and dicyclic groups G. We establish that Θ (G) admits a (k, 1) -partition for cyclic, dihedral, and dicyclic groups G of specified orders.

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Cite This Study

Ranjan et al. (Mon,) studied this question.

synapsesocial.com/papers/68d473a631b076d99fa6bea7 https://doi.org/https://doi.org/10.48550/arxiv.2507.15993