Dan Spielman (Yale University)
Abstract. We will show how physical metaphors can help us understand the structure of a graph. The graphs arising in different disciplines can have very different characteristics: social networks, protein-protein interactions networks, road networks, and scientific meshes are all graphs. But, they can look very different from each other. This diversity makes it difficult to understand arbitrary graphs.
We will explore an approach to understanding graphs that has been unreasonably successful: imagining that a graph represents a physical object. For example, we may pretend that the edges of a graph are springs, rubber bands, or resistors. Linear algebraic techniques for understanding these physical systems naturally lead to the development of spectral and algebraic graph theory. We will survey some of the fundamental ideas from these fields.