{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/weighted-theta-functions-for-non-commutative","title":"Weighted theta functions for non-commutative graphs","arxiv_id":"2101.00162","date":"2021-01-01","proceeding":null,"authors":["Dan Stahlke"],"abstract":"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.","url_abs":"https://arxiv.org/abs/2101.00162v1","url_pdf":"https://arxiv.org/pdf/2101.00162v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"weighted-theta-functions-for-non-commutative","repo_url":"https://github.com/dstahlke/NoncommutativeGraphs.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}