<p>In this present paper, we introduce graphical Branciari $ \aleph $-metric space and prove the fixed point theorem for $ \varOmega $-$ \mathcal{Q} $-contraction on complete graphical Branciari $ \aleph $ -metric spaces. Our result has been supplemented with suitable, non trivial examples. We have applied the derived fixed point result to solve non-linear Fredholm integral equations and fractional differential equation.</p>