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We present electronic structure calculations based on a single-parameter plane wave expansion method for basic graphene building blocks, namely n-oligophenylenes and n-oligoacenes, revealing excellent agreement with density-functional theory. When oligophenylene molecules are joined through meta (zigzag) or ortho (chevron) junctions, the resulting molecular dimers and polymers exhibit a semiconducting character. While zigzag dimers of oligoacenes also exhibit gapped electronic structures, their chevron-phase features a sharp metallic band at the Fermi energy. This zero-point-energy state, which transforms into Dirac-like band in chevron polymers, survives at the outer elbows of the dimer irrespective of the molecular length, and has the same origin as reported for the polyacetylene and topologically induced edge states at edge-decorated graphene nanoribbons. These findings assist the engineering of topological electronic states at the molecular level and complement the toolbox of quantum phases in carbon-based nanostructures., JEO acknowledges financial support from the Spanish Ministry of Science and Innovation (Grants PID2019-107338RB-C63) and the Basque Government (Grant IT-1255-19).