拓扑形变下标记图的计数及应用

In 1930's, the well-known Redfield-Pólya's enumeration theory was developed while studying the molecular graph enumeration problem. The theory is considered as a classical result in mathematics and has been extensively used by chemists to count the chemical isomers. Stimulated by the enumeration problem of the rigid stereo molecules in three dimensional Euclidean space, Redfield-Pólya's theory was extended for studying some further properties of molecules such as the chirality-detecting. One of these works is known as the USCI enumeration theory which was summarized by Fujita. In the past decades, various types of the large-scale topological molecules have been synthesized with the fast growing of the molecular synthesis technology such as the DNA self-assembly techniques. The non-rigid character of these topological molecules have received lots of attention from chemists and mathematicians. Form the view point of mathematics, a topological molecule may be modeled as an embedded graph in 3-dimmentional space, which is essentially distinct from the geometric character what a rigid small molecule appears in Euclidean space. This project will focus on the enumeration problem of the marked graphs under the topological deformation. In addition to its own application in chemistry, the project has the following theoretical significances: (1). The main study opens a new research subject where the topology theory and the classic enumeration theory are combined. This is an innovative idea and little appears in the current literatures along the same line；(2). The symmetry and chirality of a geometric object in Euclidean space are characterized by the rotations and the mirror reflections which leave the object invariant while, for a topological object, are by the space deformations which leave the object invariant. Therefore, the project is expected to significantly enrich the traditional symmetry group-based enumeration theory.

上世纪30年代,由图的自同构群发展出了经典的Redfield-Pólya计数理论,而三维欧氏空间中刚性立体分子的计数问题则进一步促进了该理论的完善，如Fujita的USCI理论。近几十年来，大分子合成技术的日益成熟使各种拓扑分子被人工合成，这种分子的非"刚性"引起了化学家的格外关注。从数学的角度，拓扑分子可抽象为一个有标记的拓扑嵌入图，这与小分子在欧氏空间所呈现的刚性特征有本质的区别。本项目将研究拓扑形变意义下嵌入标记图的计数问题，除应用价值外，理论意义在于：1.研究的内容是拓扑与经典的组合计数理论相结合的一个新课题；2.欧氏几何体的对称与手性是建立在刚体旋转及镜面反射基础上的，而拓扑嵌入图的对称与手性则是空间连续形变意义下的。因此，项目的实施将对传统的以对称群为基础的计数理论有所丰富和发展.

项目按照计划书及调整后的研究内容实施。调整后增加了（标记）图及链环着色模式的计数问题，方法上增加了对容斥计数理论的探索和应用。项目获得了一系列较好的研究成果，基本完成了既定的研究目标，部分研究内容取得了突破性的进展。主要包括：1. 本质上改进了容斥对消理论，即：首次提出了不依赖指标序（index order）的容斥对消方法，发展了经典的组合计数容斥原理，为容斥形式的计数问题提供了更好的理论工具，具有广泛的应用价值。2. 运用容斥对消理论对一些复杂的离散结构的计数进行估值分析，并在图着色的计数问题上取得了较大的进展，大幅改进了 Donner 和 Thomassen 的重要结果。 3.在反映大分子图某些结构特征（如蛋白质链的折叠程度）的 Estrada 指标的研究上取得了较好的成果，成果在发表后的三年时间里收到了广泛的关注，被 SCI 期刊论文引用 12 次。4. 将纽结的 Fox p-染色计数理论推广到行列式不为 0 的多分支链环上，指出Fox p-染色的最佳对象是行列式不为 0 的链环，丰富了 Fox p-染色计数理论。5. 在其它方面如图的列表着色、Pfaffian 定向及图的控制集等也取得了一系列进展。项目成果为后续标记图的计数及更一般的组合计数问题提供了良好的理论工具和思想方法。