  We address the problem of identifying families of discrete models naturally
flowing in continuum limit to relativistic quantum field theories. We call them
Dirac graphs. In this work, we require the graphs to obey spectrality property,
which implies that the adjacency matrix can be represented by a continuous
function defined on the spectral space. We also consider deformations of such
graphs away from the spectrality. We show that deformations with regular
continuum limit result in background gauge field and gravity. Interestingly,
gauge interactions appear due to contribution microscopic deformations while
the standard gravity can only be a result of macroscopic (adiabatic)
deformations.
