Combinatorics and Graph Theory

Harmonic Graphs Conjecture: Graph-Theoretic Attributes and their Number Theoretic Correlations

Authors: Felipe Correa Rodríguez
Comments: 6 Pages. Perfected version. I posted this but time has passed already and there weren't updates on the replacement, so I fear a mistake.

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.
Category: Combinatorics and Graph Theory