Bipartite Subgraphs And The Smallest Eigenvalue

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 ∆ satisfies μ > −∆ + 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.