Flat bands of periodic graphs

We study flat bands of periodic graphs in a Euclidean space. These are infinitely degenerate eigenvalues of the corresponding adjacency matrix, with eigenvectors of compact support. We provide some optimal recipes to generate desired bands and some sufficient conditions for a graph to have flat bands, we characterize the set of flat bands whose eigenvectors occupy a single cell, and we compute the list of such bands for small cells. We next discuss the stability and rarity of flat bands in special cases. Additional folklore results are proved, and many questions are still open.