  Graphene on $L1_0$-FePd(001), which has been experimentally studied in recent
years, is a heterogeneous interface with a significant lattice symmetry
mismatch between the honeycomb structure of graphene and tetragonal alloy
surface. In this work, we report on the density functional study of its
atomic-scale configurations, electronic and magnetic properties, and adsorption
mechanism, which have not been well understood in previous experimental
studies. We propose various atomic-scale models, including simple nontwisted
and low-strain twisted interfaces, and analyze their energetical stability by
performing structural optimizations using the van der Waals interactions of
both DFT-D2 and optB86b-vdW functionals. The binding energy of the most stable
structure reached $E_\mathrm{B}=-0.22$~eV/atom for DFT-D2
($E_\mathrm{B}=-0.19$~eV/atom for optB86b-vdW). The calculated FePd-graphene
spacing distance was approximately 2~\AA, which successfully reproduced the
experimental value. We also find out characteristic behaviors: the modulation
of $\pi$-bands, the suppression of the site-dependence of adsorption energy,
and the rise of \color{blue} moir\'e-like \color{black} corrugated buckling. In
addition, our atomic structure is expected to help build low-cost computational
models for investigating the physical properties of $L1_0$
alloys/two-dimensional interfaces.
