{"title":"Connectivity gaps among matroids with the same enumerative invariants","authors":"Joseph E. Bonin, Kevin Long","doi":"10.1016/j.aam.2023.102648","DOIUrl":null,"url":null,"abstract":"<div><p>Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the <span><math><mi>G</mi></math></span>-invariant and the configuration of the matroid. We show that the same is not true of the most basic connectivity invariants. Specifically, we show that for any positive integer <em>n</em>, there are pairs of matroids that have the same configuration (and so the same <span><math><mi>G</mi></math></span>-invariant and the same Tutte polynomial) but the difference between their Tutte connectivities exceeds <em>n</em><span>, and likewise for vertical connectivity and branch-width. The examples that we use to show this, which we construct using an operation that we introduce, are transversal matroids that are also positroids.</span></p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-12-08","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/S0196885823001665","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
Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the -invariant and the configuration of the matroid. We show that the same is not true of the most basic connectivity invariants. Specifically, we show that for any positive integer n, there are pairs of matroids that have the same configuration (and so the same -invariant and the same Tutte polynomial) but the difference between their Tutte connectivities exceeds n, and likewise for vertical connectivity and branch-width. The examples that we use to show this, which we construct using an operation that we introduce, are transversal matroids that are also positroids.
矩阵的许多重要枚举不变式都可以从它的图特多项式中获得，而更多的枚举不变式是由两个更强的不变式--G不变式和矩阵的配置--决定的。我们证明，最基本的连接性不变式并非如此。具体来说，我们证明了对于任何正整数 n，都存在一对具有相同配置（因此具有相同的 G 不变式和相同的 Tutte 多项式）的矩阵，但是它们的 Tutte 连接度之间的差异超过了 n，垂直连接度和分支宽度也是如此。我们用来证明这一点的例子是横向矩阵，它们也是正多边形。
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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