, Volume 6, Issue 2, pp 83-96

First online:

Eigenvalues and expanders

  • Noga AlonAffiliated withDepartment of Mathematics, Massachusetts Inst. of TechnologyDepartment of Mathematics, Tel Aviv University

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Linear expanders have numerous applications to theoretical computer science. Here we show that a regular bipartite graph is an expanderif and only if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result, which has an analytic analogue for Riemannian manifolds enables one to generate expanders randomly and check efficiently their expanding properties. It also supplies an efficient algorithm for approximating the expanding properties of a graph. The exact determination of these properties is known to be coNP-complete.

AMS subject classification (1980)

05 C 99 05 C 50 68 E 10