This paper presents a new procedure for identifying that part of the pi electronic energy of conjugated hydrocarbons which results from cyclic circulation of electrons around a ring. It first shows that one may calculate perturbatively the ground state energy of the Huckel Hamiltonian from a strongly localized Kekule-type zero-order wave function. The contributions due to cyclic circulation of the electrons appear explicitly, in terms of the interatomic hopping integral t, at the second order in cyclobutadiene (where it is equal to -t (antiaromatic)) and at third order in benzene, where its value is 0.5t (aromatic). Conjugated isomers of benzene are also considered. The cyclic circulation contributions for an N-membered ring are shown to depend strongly on the molecular graph in which it is embedded. A general expression is found for the cyclic contribution to the n energy of a ring, the Kekule graph of which contains N double bonds alternating with N single bonds. It is the energy of the ring, plus the sum of the energies of the N subsystems that result from one double-bond removal, minus the sum of the energies of the N open systems that result from one single-bond cut. This new aromaticity index, ACE(MC), may be seen as the enthalpy of a hyperhomodesmotic chemical equation. In contrast to the index ACE(DC) previously defined from a double cut of the ring, the multiple-cut ACE(MC) exhibits the expected asymptotic disappearance of the cyclic energy as the ring size tends to infinity. In the multiple-cut approach, aromaticity persists in bond-alternating rings, but, in contrast to the total n energy, the purely cyclic contribution tends to resist distortion. Extension of the approach to charged, branched and heterosubstituted rings are discussed, as well as its ab initio transcription.