We compute the barriers and heats of reaction for the conversion of benzene to naphthalene by acetylene additions using the B3LYP/4-31G approach. The Frenklach mechanism for adding a ring to a PAH molecule is compared with the Bittner-Howard acetylene addition mechanism. Both have low barriers, and the similarities and differences of the two mechanisms are discussed. A ring addition mechanism for cations is also studied, and it is found to have no barriers.