Eigenvalues and expanders

Abstract

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.

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The research was supported by the Weizmann Fellowship for Scientific Research and by Air Force Contract OSR 82-0326.

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Alon, N. Eigenvalues and expanders. Combinatorica 6, 83–96 (1986). https://doi.org/10.1007/BF02579166

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AMS subject classification (1980)

  • 05 C 99
  • 05 C 50
  • 68 E 10