Ising spins with mixed exchange interactions on regular polyhedra

An Ising Hamiltonian with mixed ferromagnetic (F) and anti-ferromagnetic (AF) interactions is extended to the case of regular polyhedra. Spins are located at the vertices while exchange interactions or bonds take place along the edges. Equal amount and equal magnitude are assumed for the two kinds of bonds. The tetrahedron, octahedron, hexahedron (cube), icosahedron and dodecahedron are studied. The fullerene structure with 60 vertices (spins) is also included. We restrict ourselves to the following ground state properties: energy per bond, remnant entropy, degeneracy of first excited level, site spin order parameter, curved plaquettes and average frustration segment. Calculations are both analytical (for small polyhedra) and computational. A statistical analysis based on 1000 different random distributions of bonds is carried out. A comparison of these results with the equivalent two-dimensional Ising lattices is also performed.