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Weighted theta functions for non-commutative graphs

Grötschel, Lovász, and Schrijver generalized the Lovász $\vartheta$ 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 Grötschel, Lovász, 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.