Some previously unknown relationships for determining the a4 and a6 coefficients of the characteristic polynomial for polycyclic aromatic hydrocarbons are presented for the first time. The structural information contained in these coefficients is more fully revealed. The equations derived for a4 and a6 allow one to determine the characteristic polynomial by inspection for many small molecular graphs. Some relationships for a8 and a10 are presented. The set of known graphical invariants (GI) or properties that remain unchanged in isomeric PAH6s is now shown to be GI={a4, a6+n0+2r6, a8c, ds+NIc, Nc, Nh, NIc+NPc, q, r}.