DescriptionExpanders are infinite families of highly connected sparse graphs. A graph is a combinational object consisting of vertices and edges connecting these vertices. The connectedness of a regular graph can be described both in terms of a geometric invariant, the expansion rate, and in terms of the spectrum of the adjacency matrix, the so called spectral gap. Expanders have lots of applications in computer science, in the development of robust networks, in error correcting codes, in the construction of fast algorithms, just to name a few. The study of expander graphs leads quickly to many other mathematical areas, meaning that this topic allows to experience the interplay of different mathematical areas like geometric group theory, analysis, linear algebra, representation theory, number theory, random walks and discrete geometry. The connection with highly nontrivial facts in elementary number theory appears prominently in the study of Ramanujan graphs. Connections to analysis and linear algebra come into play when studying the spectral properties of the adjacency matrix associated to a graph, and this viewpoint turned out to be very fruitful and carries the name of ''algebraic graph theory'' or ''spectral graph theory''. Numerous interesting topics can be pursued in this direction.ResourcesSome recommendable references in this topic are
email: N Peyerimhoff |