Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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)

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.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

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

Comments (0)

We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.

Leave a public comment
Send a private comment to the author(s)
* All users must log in before leaving a comment
Views 0
Downloads 0
Comments 0


×
Alerts
Notify me about updates to this article or when a peer-reviewed version is published.
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here.