{"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/upper-bound-for-the-number-of-spanning","title":"Upper bound for the number of spanning forests of regular graphs","arxiv_id":"2105.06801","date":"2021-05-14","proceeding":null,"authors":["Ferenc Bencs","Péter Csikvári"],"abstract":"We show that if $G$ is a $d$--regular graph on $n$ vertices, then the number of spanning forests $F(G)$ satisfies $F(G)\\leq d^n$. The previous best bound due to Kahale and Schulman gave $(d+1/2+O(1/d))^n$. We also have the more precise conjecture that $$F(G)^{1/n}\\leq \\frac{(d-1)^{d-1}}{(d^2-2d-1)^{d/2-1}}.$$ If this conjecture is true, then the expression on the right hand side is the best possible.","url_abs":"https://arxiv.org/abs/2105.06801v3","url_pdf":"https://arxiv.org/pdf/2105.06801v3.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":"upper-bound-for-the-number-of-spanning","repo_url":"https://github.com/bencsf/upper_bound_spanning_forests","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}