{"title":"A direct proof of well-definedness for the polymatroid Tutte polynomial","authors":"Xiaxia Guan , Xian'an Jin","doi":"10.1016/j.aam.2024.102809","DOIUrl":null,"url":null,"abstract":"<div><div>For a polymatroid <em>P</em> over <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, Bernardi et al. (2022) <span><span>[1]</span></span> introduced the polymatroid Tutte polynomial <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> relying on the order <span><math><mn>1</mn><mo><</mo><mn>2</mn><mo><</mo><mo>⋯</mo><mo><</mo><mi>n</mi></math></span> of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> is equivalent to another polynomial that only depends on <em>P</em>. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102809"},"PeriodicalIF":1.0000,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0196885824001416","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
For a polymatroid P over , Bernardi et al. (2022) [1] introduced the polymatroid Tutte polynomial relying on the order of , which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that is equivalent to another polynomial that only depends on P. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial.
Bernardi 等人（2022 年）[1] 根据 [n] 的阶 1<2<⋯<n，对 [n] 上的多母题 P 提出了多母题图特多项式 TP，它将经典的图特多项式从母题推广到多母题。在本文中，与 Tutte 利用边的阶数活动对所有生成树求和所定义的 Tutte 多项式的定义良好性的原始证明类似，我们给出了多马特人 Tutte 多项式定义良好性的直接而基本的证明。
期刊介绍：
Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas.
Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.
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