  A graph $G$ is defined encapsulating the number theoretic notion of the
Fundamental Theorem of Arithmetic. We then provide a graph theoretic approach
to the fundamental results on the coprimality of two natural numbers, through
the use of an adjacency operator $\hat{\mathbf{A}}(G)$. Lastly, these results
are used to give an alternate proof to the known result that there are
infinitely many primes in the natural numbers $\mathbb{N}$.
