{"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/algebraic-and-combinatorial-stability-of","title":"Algebraic and Combinatorial Stability of Independence Polynomials in Iterated Strong Products of Cycles","arxiv_id":"2503.07910","date":"2025-03-10","proceeding":null,"authors":["Todd Hildebrant"],"abstract":"This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish precise conditions under which these polynomials factor over finite fields, highlighting modular collapses based on prime and composite cycle lengths. We demonstrate that while real-rootedness depends on cycle parity, the combinatorial structure ensures universal log-concavity and unimodality of coefficients. A toggling argument provides a combinatorial proof of unimodality, complementing algebraic methods and offering insights into polynomial stability. These findings bridge combinatorial and algebraic perspectives, contributing to graph-theoretic frameworks with implications in statistical mechanics and information theory.","url_abs":"https://arxiv.org/abs/2503.07910v1","url_pdf":"https://arxiv.org/pdf/2503.07910v1.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":"algebraic-and-combinatorial-stability-of","repo_url":"https://github.com/thildebrant/alg-symmetry-unimodal-polys","is_official":1,"mentioned_in_paper":0,"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}