π-Electron Spectrum of Oligoperylene[Ntimescal N] -- Graphene Macromolecule and Its Derivatives

We revisit the model of graphene as a $\pi$-conjugated macromolecule \cite{PRL} and show that it allows to understand the origin and describe, in many cases explicitly, the electronic structure of graphene sheets, achiral graphene ribbons and carbon tubes of any size, and also graphene tori on the basis of the exact solution to the eigenvalue problem for the respective structures. The exact expressions of dispersion relations and detectable observables for these structures are compared with the predictions of other related models demonstrating that the revisited model either suggests the exact versions of widely used approximations or shows their limited applicability.