  We study the relationship between unit-distance representations and Lovasz
theta number of graphs, originally established by Lovasz. We derive and prove
min-max theorems. This framework allows us to derive a weighted version of the
hypersphere number of a graph and a related min-max theorem. Then, we connect
to sandwich theorems via graph homomorphisms. We present and study a
generalization of the hypersphere number of a graph and the related
optimization problems. The generalized problem involves finding the smallest
ellipsoid of a given shape which contains a unit-distance representation of the
graph. We prove that arbitrary positive semidefinite forms describing the
ellipsoids yield NP-hard problems.
