  Every graph G can be embedded in a Euclidean space as a two-distance set. The
Euclidean representation number of G is the smallest dimension in which G is
representable by such an embedding. We consider spherical and J-spherical
representation numbers of G and give exact formulas for these numbers using
multiplicities of polynomials that are defined by the Caley-Menger determinant.
One of the main results of the paper are explicit formulas for the
representation numbers of the join of graphs which are obtained from W.
Kuperberg's type theorem for two-distance sets.
