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)
How to cite:
Correa, F. Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations. Preprints2023, 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
Correa, F. Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations. Preprints2023, 2023081115. https://doi.org/10.20944/preprints202308.1115.v2
APA Style
Correa, F. (2023). Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations. Preprints. https://doi.org/10.20944/preprints202308.1115.v2
Chicago/Turabian Style
Correa, F. 2023 "Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations" Preprints. 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.
Received:
17 August 2023
Commenter:
Felipe Correa
Commenter's Conflict of Interests:
Author
Comment:
Enhancements in elucidating each definition, accompanied by a succinct and streamlined presentation of proofs, facilitate accessibility for a broader audience. This refinement encompasses the incorporation of essential annotations. The discourse is enriched through the augmentation and elaboration of content, fostering a deeper comprehension and engagement. All of these changes were based in responses of other mathematicians. I would also like to mention that I am an independent researcher even if I am associated with an institution.
Commenter: Felipe Correa
Commenter's Conflict of Interests: Author