We calculate the quantum states of regular polygons made of 1D quantum wires treating each polygon vertex as a scatterer. The vertex scattering matrix is analytically obtained from the model of a circular bend of a given angle of a 2D nanowire. In the single mode limit the spectrum is classified in doublets of vanishing circulation, twofold split by the small vertex reflection, and singlets with circulation degeneracy. Simple analytic expressions of the energy eigenvalues are given. It is shown how each polygon is characterized by a specific spectrum.
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