On plane graphs with link component number equal to the nullity

设是一个图。的基圈数(nullity)定义为，这里是的连通分支数。当是连通平图时，由著名的欧拉公式知等于平图的有界面的个数。令，这里是图的Tutte多项式。当是平图时，恰好是图所对应的链环的连通分支数，我们称是图的链环分支数。易证，对于任意的连通平图有。在文[6]中，作者刻画了极大图，即的连通平图。 本文的研究工作是文[6]研究工作的延续。首先我们研究次极大图即的连通平图。证明了最小度至少是的连通平图是次极大图当且仅当同构于完全图。基于该结果，给出了判断一个连通平图是否为次极大图的一个简单算法，并证明了所有的次极大图可由，和两个图运算构造得到。 然后我们将极大图和次极大图的研究结果推广到任...Let G=(V,E) be a graph. The nullity of the graph G is defined to be , where K(G) is the number of connected components of the graph G. When G is a connected plane graph, n(G) is equal to the number of bounded faces of the plane graph G by the well-known Euler formula. Let , where be the Tutte polynomial of the graph G. We call the link component number of the graph G, since when G is a ...学位：理学硕士院系专业：数学科学学院数学与应用数学系_应用数学学号：X200717000

The Jones polynomial and graphs on surfaces

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection. We give some applications of this approach.Comment: 19 pages, 9 figures, minor change

The Interlace Polynomial

In this paper, we survey results regarding the interlace polynomial of a graph, connections to such graph polynomials as the Martin and Tutte polynomials, and generalizations to the realms of isotropic systems and delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials, edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL