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【学术讲座】图的Tutte多项式近期研究进展
2017-10-16 18:33   审核人:   (点击: )

报告题目:图的Tutte多项式近期研究进展

报告人:叶永南教授

讲座时间:2017年10月21日10:15-11:00

讲座地点:友谊校区国际会议中心第一会议室

邀请人:张胜贵

承办学院:理学院

联系人:陆由

联系电话:18202966680

报告简介

William Tutte is one of the founders of the modern graph. For every undirected graph, Tutte defined a polynomial TG(x,y) in two variables which plays an important role in graph theory. In this talk, we will introduce some recent progresses in studies of the Tutte polynomial of a graph.

In Tutte's original definitions, non-negative integers, called internal and external activities with respect to the arbitrary enumeration, are defined for each spanning tree, they serve as the indices of x and y in the product that is the corresponding term of Tutte polynomial. First, We will introduce the conceptions of \sigma-cut tail and \sigma-cycle tail of T, which are generalizations of the conceptions of internally and externally activities, repectively, where \sigma is a sequence on the edge set of G and T is a spanning tree of G. We will also discuss the conceptions of proper Tutte mapping and deletion-contraction mapping.

In 2004, Postnikov and Shapiro introduced the concept of G-parking functions in the study of certain quotients of the polynomial ring. The Tutte polynomial of the graph G can be expressed in terms of statistics of G-parking functions. Let ? be a nonsingular M-matrix. We will introduce ?-parking functions which is a generalization of G-parking functions. We will introduce the abelian sandpile model and ?-recurrent configurations. There is a simple bijection between ?-parking functions and ?-recurrent configurations. We will discuss the geometry of sandpile model.

In general, the Tutte polynomial encodes information about subgraphs of G. For example, for a connected graph G, TG(1, 1) is the number of spanning trees of G, TG(2, 1) is the number of spanning forests of G, TG(1, 2) is the number of connected spanning subgraphs of G, TG(2, 2) is the number of spanning subgraphs of $G$. At last, we will discuss combinatorial interpretations of TG(1+p, -1)$ and TG(-1, 1).

报告人简介

叶永南,台湾中研院数学研究所研究员,1985年在美国纽约州立大学水牛城分部获得博士学位,1987年7月返台担任中央研究院数学所副研究员,1991年1月晋升为研究员迄今。曾任加拿大魁北克大学蒙特娄分部资讯与数学系研究学者,麻省理工学院数学系、柏克莱加州大学统计系和澳洲Monash大学经济系访问学者。学术研究除了数学之外,还涉及物理化学、统计、经济等多个领域。曾任台湾数学推动中心主任,中研院数学所副所长,多次获得台湾中研院杰出研究奖,国科会杰出研究奖,国科会杰出研究计划奖。已发表的论文有百余篇,组合论国际顶级杂志JCTA曾出版专门文章介绍Yeh-species,这个由叶永南研究员名字命名的领域,现在这一方向的研究仍然在不断深入。目前,叶永南研究员的研究主要在图的Tutte多项式及其相关组合结构、计数组合学中uniform partitions等方面。

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