Flat-Bands on Partial Line Graphs -- Systematic Method for Generating   Flat-Band Lattice Structures
  We introduce a systematic method for constructing a class of lattice structures that we call ``partial line graphs''.In tight-binding models on partial line graphs, energy bands with flat energy dispersions emerge.This method can be applied to two- and three-dimensional systems. We show examples of partial line graphs of square and cubic lattices. The method is useful in providing a guideline for synthesizing materials with flat energy bands, since the tight-binding models on the partial line graphs provide us a large room for modification, maintaining the flat energy dispersions. 
Shin Miyahara, Kenn Kubo, Hiroshi Ono, Yoshihiro Shimomura and Nobuo   Furukawa