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The excitonic level structure of a ring-like chain of dimers is discussed analytically in order to aid the understanding of the possible spectral properties of LH1 and LH2 of purple photosynthetic bacteria. Under the approximation of dipoledipole interaction between Bchls, the excitonic levels, bandwidths and energy gap between two Davydov subbands are expressed analytically in terms of interaction energies and configurational parameters of dipoles. Our model includes all the interactions between pigment molecules in the system. The oscillator strengths and circular dichroism (CD) for the excitonic states are also presented analytically. The simulated absorption and CD spectra of LH1 and LH2 complexes reproduce the main features of the measured results.