Mathematical Sciences: Structure and Rigidity of Graphs with Applications to Network Models of Materials

9412017 Franzblau在这个项目中，首席研究人员的目标是获得以下结果：(1)实现一个实用的组合算法来计算网络(3维或更多维)的自由度界限，(2)新的刚性组合条件或图族中自由度的简单公式，(3)新的、易于计算的眼镜网络模型中的中程有序度量。所采用的方法包括组合优化、图论和离散算法设计。关于网络模型的问题主要是通过计算机实验来解决的，并利用研究人员开发和实现的算法。该项目有两个相互关联的目标，旨在为数学和材料科学贡献新的成果。第一个目标是解决刚性数学理论中的关键公开问题，包括计算网络(也称为图)的自由度。这一理论有很长的历史，其中包括麦克斯韦(1864年)的工作，他确定了由刚性杆和可移动关节组成的“脚手架”本身是否为刚性的。第二个目标是解决固体网络模型的基本问题；这种模型(也称为球棒模型)，其中点代表原子，点之间的连接代表化学键，经常被研究以更好地理解晶体固体和玻璃材料的性质。一个重点是刻画网络模型的结构与其刚性之间的关系，另一个重点是寻找有效的度量来捕捉这种网络结构。因此，这项工作有助于更好地了解材料的性质。

9412017 Franzblau In this project, the principal investigator aims to obtain the following results: (1) an implementation of a practical combinatorial algorithm to compute bounds on degrees of freedom of a network (in 3 or more dimensions), (2) new combinatorial conditions for rigidity or simple formulas for degrees of freedom in families of graphs, (3) new, easily computed measures of medium-range order in network models of glasses. The methods employed include those of combinatorial optimization, graph theory, and discrete algorithm design. Problems on network models are addressed largely through computer experiments, and make use of algorithms developed and implemented by the investigator. This project has two related aims, and is intended to contribute new results both to mathematics and materials science. The first aim is to address key open problems in the mathematical theory of rigidity, including computing the degrees of freedom of a network (also called a graph). This theory has a long history, which includes the work of Maxwell (1864) on determining whether a "scaffold" made of rigid bars and movable joints is itself rigid. The second aim is to address basic issues on network models of solids; such models (also called ball-and-stick models), in which points represent atoms and connections between points represent chemical bonds, are often studied to better understand the properties of both crystalline solids and glassy materials. One focus is to characterize the relationship between the structure of a network model and its rigidity, and the other is to find useful measures which capture this network structure. The work therefore leads to a better understanding of materials properties.