{"title":"论随机多边形链中完全匹配的数量","authors":"Shouliu Wei, Yongde Feng, Xiaoling Ke, Jianwu Huang","doi":"10.1515/math-2023-0146","DOIUrl":null,"url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0146_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a graph. A perfect matching of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0146_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a regular spanning subgraph of degree one. Enumeration of perfect matchings of a (molecule) graph is interest in chemistry, physics, and mathematics. But the enumeration problem of perfect matchings for general graphs (even in bipartite graphs) is non-deterministic polynomial (NP)-hard. Xiao et al. [C. Xiao, H. Chen, L. Liu, <jats:italic>Perfect matchings in random pentagonal chains</jats:italic>, J. Math. Chem. 55 (2017), 1878–1886] have studied the problem of perfect matchings for random odd-polygonal chain (i.e., with odd polygons). In this article, we further present simple counting formulae for the expected value of the number of perfect matchings in random even-polygonal chains (i.e., with even polygons). Based on these formulae, we obtain the average values of the number for perfect matchings with respect to the set of all even-polygonal chains with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0146_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>n</m:mi> </m:math> <jats:tex-math>n</jats:tex-math> </jats:alternatives> </jats:inline-formula> polygons.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"109 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the number of perfect matchings in random polygonal chains\",\"authors\":\"Shouliu Wei, Yongde Feng, Xiaoling Ke, Jianwu Huang\",\"doi\":\"10.1515/math-2023-0146\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0146_eq_001.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a graph. A perfect matching of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0146_eq_002.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a regular spanning subgraph of degree one. Enumeration of perfect matchings of a (molecule) graph is interest in chemistry, physics, and mathematics. But the enumeration problem of perfect matchings for general graphs (even in bipartite graphs) is non-deterministic polynomial (NP)-hard. Xiao et al. [C. Xiao, H. Chen, L. Liu, <jats:italic>Perfect matchings in random pentagonal chains</jats:italic>, J. Math. Chem. 55 (2017), 1878–1886] have studied the problem of perfect matchings for random odd-polygonal chain (i.e., with odd polygons). In this article, we further present simple counting formulae for the expected value of the number of perfect matchings in random even-polygonal chains (i.e., with even polygons). Based on these formulae, we obtain the average values of the number for perfect matchings with respect to the set of all even-polygonal chains with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_math-2023-0146_eq_003.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>n</m:mi> </m:math> <jats:tex-math>n</jats:tex-math> </jats:alternatives> </jats:inline-formula> polygons.\",\"PeriodicalId\":48713,\"journal\":{\"name\":\"Open Mathematics\",\"volume\":\"109 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-12-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Open Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/math-2023-0146\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Open Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/math-2023-0146","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设 G G 是一个图。G G 的完美匹配是阶数为 1 的正则遍历子图。枚举（分子）图的完全匹配是化学、物理和数学领域的兴趣所在。但对于一般图（即使是二方图）来说，完美匹配的枚举问题是非确定性多项式（NP）困难的。Xiao et al.Xiao, H. Chen, L. Liu, Perfect matchings in random pentagonal chains, J. Math.Chem.55 (2017), 1878-1886] 研究了随机奇多边形链（即奇多边形）的完全匹配问题。在本文中，我们进一步提出了随机偶多边形链（即偶数多边形）中完全匹配次数期望值的简单计数公式。根据这些计算公式，我们可以得到具有 n n 个多边形的所有偶多边形链的完全匹配数的平均值。
On the number of perfect matchings in random polygonal chains
Let GG be a graph. A perfect matching of GG is a regular spanning subgraph of degree one. Enumeration of perfect matchings of a (molecule) graph is interest in chemistry, physics, and mathematics. But the enumeration problem of perfect matchings for general graphs (even in bipartite graphs) is non-deterministic polynomial (NP)-hard. Xiao et al. [C. Xiao, H. Chen, L. Liu, Perfect matchings in random pentagonal chains, J. Math. Chem. 55 (2017), 1878–1886] have studied the problem of perfect matchings for random odd-polygonal chain (i.e., with odd polygons). In this article, we further present simple counting formulae for the expected value of the number of perfect matchings in random even-polygonal chains (i.e., with even polygons). Based on these formulae, we obtain the average values of the number for perfect matchings with respect to the set of all even-polygonal chains with nn polygons.
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Open Mathematics - formerly Central European Journal of Mathematics
Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant, original and relevant works in all areas of mathematics. The journal provides the readers with free, instant, and permanent access to all content worldwide; and the authors with extensive promotion of published articles, long-time preservation, language-correction services, no space constraints and immediate publication.
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