  Motivated by questions in algebra and combinatorics we study two ideals
associated to a simple graph G:
  --> the Lovasz-Saks-Schrijver ideal defining the d-dimensional orthogonal
representations of the graph complementary to G and
  --> the determinantal ideal of the (d+1)-minors of a generic symmetric with
0s in positions prescribed by the graph G.
  In characteristic 0 these two ideals turns out to be closely related and
algebraic properties such as being radical, prime or a complete intersection
transfer from the Lovasz-Saks-Schrijver ideal to the determinantal ideal. For
Lovasz-Saks-Schrijver ideals we link these properties to combinatorial
properties of G and show that they always hold for d large enough. For specific
classes of graph, such a forests, we can give a complete picture and classify
the radical, prime and complete intersection Lovasz-Saks-Schrijver ideals.
