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This is the first of two papers devoted to tight-binding electronic spectraon graphs with the topology of the sphere. In this work the one-electronspectrum is investigated as a function of the radial magnetic field produced bya magnetic charge sitting at the center of the sphere. The latter is an integermultiple of the quantized magnetic charge of the Dirac monopole, that integerdefining the gauge sector. An analysis of the spectrum is carried out for thefive Platonic solids (tetrahedron, cube, octahedron, dodecahedron andicosahedron), the C$_{60}$ fullerene, and two families of polyhedra, thediamonds and the prisms. Except for the fullerene, all the spectra are obtainedin closed form. They exhibit a rich pattern of degeneracies. The total energyat half filling is also evaluated in all the examples as a function of themagnetic charge.

Tight-binding electronic spectra on graphs with spherical topology. I. The effect of a magnetic charge