Dispersion of "Dispersionless Zero Mode": Comments on L. Brey and H.A.   Fertig paper Electronic States of Graphene Nanoribbons Studied With The Dirac   Equation
  As a particular application of the earlier proposed model of graphene as a macromolecule, we found the exact analytical expression of dispersion relation for the band of edge states in graphene zigzag ribbons. This band is often referred to as "dispersionless band" or "zero mode". The obtained result contrasts description of edge states given in the referenced paper, showing that the earlier given explanation is valid only for a very narrow region of values of the electron/hole wave vector, but for the rest, it is not correct. 
Lyuba Malysheva and Alexander Onipko