  This work explores the possibility to transfer the parity law of the
singlet-triplet gap established for square ladders (gapped for even number of
legs, gapless for odd number of legs) to fused polyacenic 1-D systems, i.e.,
graphite ribbons. Qualitative arguments are presented in favor of a gapped
character when the number $n\_{\omega}$ of benzene rings along the ribbon width
is odd. A series of numerical calculations (quantitative mapping on spin 1/2
chains, renormalized excitonic treatments and Quantum Monte Carlo) confirm the
parity law and the gapless character of the ribbon for even $n\_{\omega}$.
