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A graph $G=(V,E)$ is called an expander if every vertex subset $U$ of size upto $|V|/2$ has an external neighborhood whose size is comparable to $|U|$.Expanders have been a subject of intensive research for more than three decadesand have become one of the central notions of modern graph theory.  We first discuss the above definition of an expander and its alternatives.Then we present examples of families of expanding graphs and state basicproperties of expanders. Next, we introduce a way to argue that a given graphcontains a large expanding subgraph. Finally we research properties ofexpanding graphs, such as existence of small separators, of cycles (includingcycle lengths), and embedding of large minors.

Expanders - how to find them, and what to find in them