Fulvene is a non-aromatic molecule, but variation of the electron-donating/withdrawing power of substituents exo to the five-membered ring can drive the system between the extremes of aromatic and antiaromatic, as judged by prediction of fully developed diatropic and paratropic ring currents through ab initio calculations made at the ipsocentric 6-31G**/CTOCD-DZ CHF level.

Conditions for transmission of a $\pi$ -conjugated molecular conductor are derived within the source and sink potential approach in terms of numbers of nonbonding levels of four graphs: The molecular graph $G$ and the three vertex-deleted subgraphs obtained by removing one or both contact vertices. For all bipartite and most nonbipartite $G$ , counting nonbonding levels gives a simple necessary and sufficient condition for conduction at the Fermi level. The exceptional case is where $G$ is nonbipartite and all four graphs have the same number of nonbonding levels; then, an auxiliary requirement involving tail coefficients of the four characteristic polynomials must also be checked.

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 Kekulé 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.