  We present a thorough tight-binding analysis of the band structure of a wide
variety of lattices belonging to the class of honeycomb and Kagome systems
including several mixed forms combining both lattices. The band structure of
these systems are made of a combination of dispersive and flat bands. The
dispersive bands possess Dirac cones (linear dispersion) at the six corners (K
points) of the Brillouin zone although in peculiar cases Dirac cones at the
center of the zone $(\Gamma$ point) appear. The flat bands can be of different
nature. Most of them are tangent to the dispersive bands at the center of the
zone but some, for symmetry reasons, do not hybridize with other states. The
objective of our work is to provide an analysis of a wide class of so-called
ligand-decorated honeycomb Kagome lattices that are observed in 2D
metal-organic framework (MOF) where the ligand occupy honeycomb sites and the
metallic atoms the Kagome sites. We show that the $p_x$-$p_y$ graphene model is
relevant in these systems and there exists four types of flat bands: Kagome
flat (singly degenerate) bands, two kinds of ligand-centered flat bands (A$_2$
like and E like, respectively doubly and singly degenerate) and metal-centered
(three fold degenerate) flat bands.
