An Analytical Illustration of the Relevance of Molecular Topology to the Auifbau Process*

The ideas of Mallion and Rouvray (1978), concerning the relevance of molecular topology to the prospects of obtaining, on the basis of the Aufbau Principle, a unique, n:-electronic ground-state configuration for an existent or hypothetical conjugated-system, are extended by considering a series of networks introduced by Balaban in 1978. It is shown by exploiting the properties of the eigenvalues of circulant matrices that the graph spectrum of a general member of this series may be found analytically in closed form. From this it is further deduced that application of the Aufbau process to the »Balaban graphs«, BN, will lead to the establishment of a unique, ground-state configuration if, and only if, N is divisible by 4. The Balaban graphs are thus shown to constitute a series in which networks that give rise to a unique, ground-state configuration when the Aufbau Principle is invoked alternate with ones that do not. As a result of these observations, it is emphasised that, despite what is often assumed to the contrary, the existence of a unique and unambiguous »n-electronic, ground-state configuration« for an arbitrary network should not be taken for granted.