It is shown that, within the tight-binding approximation, Fermi-level ballistic conduction for a perimeter-connected graphene fragment follows a simple selection rule: the zero eigenvalues of the molecular graph and of its subgraph minus both contact vertices must be equal in number, as must those of the two subgraphs with single contact vertices deleted. In chemical terms, the new rule therefore involves counting nonbonding orbitals of four molecules. The rule is initially derived within the source and sink potential scattering framework, but has equivalent forms that unify the molecular-orbital and valence-bond approaches to conduction. It is shown that the new selection rule can be cast in terms of Kekule counts, bond orders, and frontier-orbital coefficients. In particular, for a Kekulean graphene, conduction pathways are shown to be ranked in efficiency by a (nonmonotonic) function of Pauling bond order between the contact vertices. Frontier-orbital analysis of conduction approximates this function. For a monoradical graphene, the analogous function is shown to depend on Pauling spin densities at contact vertices.