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.v1
Correa, F. Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations. Preprints 2023, 2023081115. https://doi.org/10.20944/preprints202308.1115.v1
Correa, F. Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations. Preprints2023, 2023081115. https://doi.org/10.20944/preprints202308.1115.v1
APA Style
Correa, F. (2023). Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations. Preprints. https://doi.org/10.20944/preprints202308.1115.v1
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.v1
Abstract
The Harmonic Graphs Conjecture states that there exists a harmonious relationship between the graph's Harmonic Index ($HI(G_n)$) and the number of vertices ($n$) for every connected graph $G_n$. This relationship can be expressed as a formula, which takes into account the prime number theorem and the sum of divisors function. In this paper, we prove the Harmonic Graphs Conjecture for cycle graphs and complete graphs. We do this by expanding the definitions of the harmonic index and the sum of divisors function, and then using the prime number theorem to approximate the values of these functions. This work is an effort to provide a contribution to the field of graph theory. It provides a new way to study the connectivity of graphs and opens up new avenues for research. For example, our results could be used to develop new algorithms for finding connected components in graphs, or to design new networks that are more resilient to failures.
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.
