The Hckel band structure is analytically investigated for a family of arbitrary-width -network polymers which may be viewed as being cut from the graphite lattice. For wider strips a portion of two nearly nonbonding bands on either side of the Fermi energy are found to be localized to opposite edges of the strip. Novel consequences of these edge-localized band orbitals are considered and correlated to a simple resonating valence bond picture. 1994.