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. Applied 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.