  Topological flat bands, such as the band in twisted bilayer graphene, are
becoming a promising platform to study topics such as correlation physics,
superconductivity, and transport. In this work, we introduce a generic approach
to construct two-dimensional (2D) topological quasi-flat bands from line graphs
and split graphs of bipartite lattices. A line graph or split graph of a
bipartite lattice exhibits a set of flat bands and a set of dispersive bands.
The flat band connects to the dispersive bands through a degenerate state at
some momentum. We find that, with spin-orbit coupling (SOC), the flat band
becomes quasi-flat and gapped from the dispersive bands. By studying a series
of specific line graphs and split graphs of bipartite lattices, we find that
(i) if the flat band (without SOC) has inversion or $C_2$ symmetry and is
non-degenerate, then the resulting quasi-flat band must be topologically
nontrivial, and (ii) if the flat band (without SOC) is degenerate, then there
exists an SOC potential such that the resulting quasi-flat band is
topologically nontrivial. This generic mechanism serves as a paradigm for
finding topological quasi-flat bands in 2D crystalline materials and
meta-materials.
