The aim of this article is to obtain common fixed point results for generalized contractions involving control functions of two variables in the context of $\mathcal{F}$-metric spaces. We also furnish an example to show the originality of our main result. Some results in the context of $\mathcal{F}$-metric space equipped with a directed graph $G$ are also established. As an application, we discuss the existence of solution to nonlinear neutral differential equation.