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Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations
Version 1
: Received: 14 August 2023 / Approved: 14 August 2023 / Online: 15 August 2023 (09:31:15 CEST)
Version 2 : Received: 16 August 2023 / Approved: 16 August 2023 / Online: 17 August 2023 (10:07:06 CEST)
Version 2 : Received: 16 August 2023 / Approved: 16 August 2023 / Online: 17 August 2023 (10:07:06 CEST)
How to cite: Correa, F. Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations. Preprints 2023, 2023081115. https://doi.org/10.20944/preprints202308.1115.v2 Correa, F. Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations. Preprints 2023, 2023081115. https://doi.org/10.20944/preprints202308.1115.v2
Abstract
The Harmonic Graphs Conjecture states that there exists an asymptotic relation involving the Harmonic Index and the natural logarithm as the order of the graph increases. This conjecture, grounded in the novel context of Prime Graphs, draws upon the Prime Number Theorem and the sum of divisors function to unveil a compelling asymptotic connection. By carefully expanding the definitions of the harmonic index and the sum of divisors function, and leveraging the prime number theorem's approximations, we establish a formula that captures this intricate relationship. This work is an effort to contribute to the advancement of graph theory, introducing a fresh lens through which graph connectivity can be explored. The synthesis of prime numbers and graph properties not only deepens our understanding of structural complexity but also paves the way for innovative research directions.
Keywords
Graph Theory, Number Theory, Primes, Conjecture, Harmonic Index
Subject
Computer Science and Mathematics, Discrete Mathematics and Combinatorics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Commenter: Felipe Correa
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