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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.