Approaching Number Theory conjectures through Graph Theory i try to find if there was an attempt to prove any famous Number Theory conjectures like Goldbach conjecture or Twin Prime conjectures through Graph Theory.
I have in my mind something like the approach to Fermat Theorem by Andrew Wiles through the Taniyama-Shimura conjecture. 
Is there something like that (succesfull or not) using Graph Theory to attack Number Theory?
There are some results proven by graph theory, however I am not sure whether those results are good examples you want. Because number theory has offsprings by cross-breeding with some other fields, like arithmetic combinatorics and additive number theory 

In number theory, Szemerédi's theorem that was formerly the Erdős–Turán conjecture is a good example proven by graph theory. (Although it is not necessary to prove by graph theory. Other proofs are using different approaches, e.g., Hillel Furstenberg uses ergodic theory, and Timothy Gowers uses both Fourier analysis and combinatorics)

And this result leads to Green–Tao theorem.