Bipartite Subgraphs and the Smallest Eigenvalue

Authors:
Noga Alon;Benny Sudakov
Affiliations:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel and Institute for Advanced Study, Princeton, NJ 08540, USA (e-mail: noga@mat ...;Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel (e-mail: sudakov@math.tau.ac.il)
Venue:
Combinatorics, Probability and Computing
Year:
2000

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Abstract

Two results dealing with the relation between the smallest eigenvalue of a graph and its bipartite subgraphs are obtained. The first result is that the smallest eigenvalue μ of any non-bipartite graph on n vertices with diameter D and maximum degree Δ satisfies μ ≥ −Δ + 1/(D+1)n. This improves previous estimates and is tight up to a constant factor. The second result is the determination of the precise approximation guarantee of the MAX CUT algorithm of Goemans and Williamson for graphs G = (V, E) in which the size of the max cut is at least A∣E∣, for all A between 0.845 and 1. This extends a result of Karloff.