To answer this question often one of the following methods, which can easily be formalised , is used:

choose N points on the sphere | choose N points on the sphere |

to construct a polyhedron (e.g. convex hull,tangential) |
calculate the distance of each pair of points |

evaluate it by its facets or edges. | evaluate these distancies |

Here you can find background for searching. Also you find some applets to:

find good point-arrangements for: |
construct point-sets on the sphere using: |
view some results |

well known problems | complete list of simple polyhedra (8 ≤ N ≤ 21) | applet |

free defined problems | different construction methods | list from literature |

point-to-point-distancies | icosahedral symmetry | obtained results |