彼得森对图因子理论的贡献

发布时间：2018-06-29 20:10

本文选题：图论 + 彼得森　；参考：《河北师范大学》2015年硕士论文

【摘要】：图论自1736年诞生之日起,在200多年的发展进程中,数学家们围绕著名的“四色问题”展开研究。20世纪以来,随着图论与其他学科的相互融合,涌现了诸如代数图论、概率图论、随机图论等众多新的学科交叉分支。图因子理论就是代数图论的范畴,主要研究一般图具有图因子的图特征问题。丹麦数学家彼得森的独创性思维和学科交叉思想致使他创作出第一篇包含图论方面(图因子理论)基本结论的文章《正则图理论》,其在图因子理论的工作被认为是具有开创性的。本文在整理、研读、分析文献的基础上,结合图因子理论发展历史,以彼得森三篇图论文章为核心展开具体研究,系统分析了彼得森解决问题的创新思维和学科交叉思想。全文主要讨论了以下问题:1.详细介绍了彼得森的生平。彼得森是19世纪下半叶丹麦数学的领军人物之一,虽生活艰苦,但不忘数学研究。独特的创造性和优雅的教学态度,使其一生受到尊重,享有国际荣誉。2.概述了图因子理论形成的学术背景。(1)泰特对“四色问题”的研究使得1-因子凸显出来。(2)西尔维斯特将形象直观的“图”应用于不变量理论的研究中,其方法为彼得森提供了创造性思维的来源。3.深究了彼得森在图因子理论的工作。以彼得森的三篇图论文章为核心展开细致研究,分析了《正则图理论》的创作初衷、研究框架及主要结论。重点剖析了彼得森的两个定理,简要分析了作为泰特“定理”反例给出的彼得森图以及彼得森在3-正则图的工作。4.阐述了彼得森对图因子理论后续发展的影响。回溯并分析了受彼得森工作影响的柯尼希、门格尔、霍尔、塔特在图因子理论的重要工作,揭示彼得森的工作在图因子理论后续发展的重要影响,进而更加明确了彼得森工作的开创性地位。
[Abstract]:Since the birth of graph theory in 1736, in the course of more than 200 years' development, mathematicians have studied the famous "four-color problem" since the 20th century. With the integration of graph theory and other disciplines, such as algebraic graph theory and probabilistic graph theory have emerged. Random graph theory and many other new interdisciplinary branches. The theory of graph factor is the category of algebraic graph theory. The Danish mathematician Peterson's original thinking and interdisciplinary thinking led him to write the first article "regular Graph Theory", which contains the basic conclusions of graph theory (graph factor theory), and its work in graph factor theory is considered to be Is groundbreaking. On the basis of sorting out, studying and analyzing the literature, combined with the history of the development of graph factor theory and taking Peterson's three graph theory articles as the core, this paper systematically analyzes Peterson's innovative thinking and interdisciplinary thinking on problem-solving. This paper mainly discusses the following question: 1. The life of Peterson was introduced in detail. Peterson was one of the leaders of Danish mathematics in the second half of the 19 th century. Unique creative and elegant teaching attitude, so that its life is respected, enjoy international honor. 2. This paper summarizes the academic background of the formation of graph factor theory. (1) Tate's study of "four-color problem" makes 1- factor prominent. (2) Sylvester applies image "graph" to the study of invariant theory. The method provides Peterson with a source of creative thinking. In this paper, Peterson's work on graph factor theory is studied. Taking Peterson's three graph theory articles as the core, the author analyzes the original intention, the research frame and the main conclusions of the regular Graph Theory. In this paper, two Peterson's theorems are analyzed, and the Peterson graphs given as a counter example of Tate's Theorem and the work of Peterson's 3-regular graph. The influence of Peterson on the following development of graph factor theory is expounded. This paper reviews and analyzes the important work of Koenig, Mengar, Hall and Tate in graph factor theory, which is influenced by Peterson's work, and reveals the important influence of Peterson's work on the subsequent development of graph factor theory. Thus, the groundbreaking position of Peterson's work was more clearly defined.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O157.5

【参考文献】