This paper by Noga Alon discusses the relationship between eigenvalues of graphs and their expanding properties. It shows that a regular bipartite graph is an expander if and only if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result has an analytic analogue for Riemannian manifolds and enables the generation of expanders randomly and the efficient checking of their expanding properties. It also provides an efficient algorithm for approximating the expanding properties of a graph. The exact determination of these properties is known to be coNP-complete. The paper introduces the concept of expanders and magnifiers, and establishes a close relationship between eigenvalues and these properties. It proves that a graph is an expander if and only if its second smallest eigenvalue is well separated from zero. This result is used to generate expanders randomly and to check their expanding properties efficiently. The paper also shows how these methods can be used to generate graphs with higher expansion properties from expanders with weaker expansion properties. It discusses the use of eigenvalues in generating expanders and magnifiers, and provides an efficient algorithm for computing eigenvalues of matrices. The paper concludes with some concluding remarks and open problems.This paper by Noga Alon discusses the relationship between eigenvalues of graphs and their expanding properties. It shows that a regular bipartite graph is an expander if and only if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result has an analytic analogue for Riemannian manifolds and enables the generation of expanders randomly and the efficient checking of their expanding properties. It also provides an efficient algorithm for approximating the expanding properties of a graph. The exact determination of these properties is known to be coNP-complete. The paper introduces the concept of expanders and magnifiers, and establishes a close relationship between eigenvalues and these properties. It proves that a graph is an expander if and only if its second smallest eigenvalue is well separated from zero. This result is used to generate expanders randomly and to check their expanding properties efficiently. The paper also shows how these methods can be used to generate graphs with higher expansion properties from expanders with weaker expansion properties. It discusses the use of eigenvalues in generating expanders and magnifiers, and provides an efficient algorithm for computing eigenvalues of matrices. The paper concludes with some concluding remarks and open problems.