Expander Graphs and their Applications Draft - not for distribution Shlomo Hoory Nathan Linial Avi Wigderson University of British Columbia The Hebrew University Institue of Advanced Study Vancouver, BC Jerusalem, Israel Princeton, NJ
[email protected] [email protected] [email protected] January 1, 2006 DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- An Overview A major consideration we had in writing this survey was to make it accessible to mathematicians as well as computer scientists, since expander graphs, the protagonists of our story come up in numerous and often surprising contexts in both fields. A glossary of some basic terms and facts from computer science can be found at the end of this article. But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In Mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.