Spectra of toroidal graphs

An n -fold periodic locally finite graph in the Euclidean n -space may be considered the parent of an infinite class of n -dimensional toroidal finite graphs. An elementary method is developed that allows the characteristic polynomials of these graphs to be factored, in a uniform manner, into smaller polynomials, all of the same size. d to the hexagonal tessellation of the plane (the graphite sheet), this method enables the spectra and corresponding orthonormal eigenvector systems for all toroidal fullerenes and (3, 6)-cages to be explicitly calculated. In particular, a conjecture of P.W. Fowler on the spectra of (3, 6)-cages is proved.