Prime labelling of centerless double wheel graph using Python programming
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Date
2024-11-01
Authors
Herath, H. M. O. D.
Weerarathna, M. D. M. C. P.
Perera, A. A. I.
Journal Title
Journal ISSN
Volume Title
Publisher
Postgraduate Institute of Science (PGIS), University of Peradeniya, Sri Lanka
Abstract
This research investigated the prime labelling of the Centerless Double Wheel graph (𝐶𝐷W₂𝑛 ) using Python programming. Prime labelling is a type of graph labelling that assigns labels to graph vertices such that the greatest common divisor of the labels of adjacent vertices is one. While most previous works on prime labelling have largely relied on manual methods to assign values to vertices based on specific conditions, this study automates the labelling process through programming. The Centerless Double Wheel graph is a simple graph obtained by using the Cartesian product of Cyclic graph; C𝑛 with 𝑛 vertices and Complete graph of the form k₂ and is denoted by 𝐶𝐷W₂𝑛 (i.e. 𝐶𝑛×k₂=𝐶𝐷𝑊₂𝑛). If 𝑛 is an even integer, then 𝐶𝐷𝑊₂𝑛 achieve its prime labelling based on whether 𝑛+1 is prime, or, 2𝑛+1 is prime. Contrarily, this study established that no prime labelling exists for 𝐶𝐷𝑊₂𝑛 when 𝑛 is odd. The proof of the theorem is given as a combinatorial version, and the algorithmic approach was detailed using Python, offering a step-by-step procedure for generating and labelling 𝐶𝐷𝑊₂𝑛 graphs. This work advances the understanding of prime graph labelling and provides computational tools to simplify the process, setting a foundation for future exploration of more complex graph structures.
Description
Keywords
Centerless Double Wheel graph , Prime labelling , Python programming
Citation
Proceedings of the Postgraduate Institute of Science Research Congress (RESCON) -2024, University of Peradeniya, P 49