  Gr\"otschel, Lov\'asz, and Schrijver generalized the Lov\'asz $\vartheta$
function by allowing a weight for each vertex. We provide a similar
generalization of Duan, Severini, and Winter's $\tilde{\vartheta}$ on
non-commutative graphs. While the classical theory involves a weight vector
assigning a non-negative weight to each vertex, the non-commutative theory uses
a positive semidefinite weight matrix. The classical theory is recovered in the
case of diagonal weight matrices.
  Most of Gr\"otschel, Lov\'asz, and Schrijver's results generalize to
non-commutative graphs. In particular, we generalize the inequality
$\vartheta(G, w) \vartheta(\overline{G}, x) \ge \langle w, x \rangle$ with some
modification needed due to non-commutative graphs having a richer notion of
complementation. Similar to the classical case, facets of the theta body
correspond to cliques and if the theta body anti-blocker is finitely generated
then it is equal to the non-commutative generalization of the clique polytope.
  We propose two definitions for non-commutative perfect graphs, equivalent for
classical graphs but inequivalent for non-commutative graphs.
