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

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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.