Exact analytical formulae for mean coordination numbers in clusters

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Rigorous analytical formulae for mean nearest-neighbour coordination numbers in clusters as a function of cluster size have been derived for a range of geometries: the tetrahedron, octahedron, cuboctahedron, icosahedron and bcc rhombic dodecahedron. Formulae for outer-neighbour coordination numbers are also reported, including a complete analysis of interatomic distances and mean coordination numbers in icosahedra. The formulae will find application in studies of electronic structure and interpretation of EXAFS data.