Eigenvalue Bounds, Spectral Partitioning, and Metrical Deformations via Flows

We present a new method for upper bounding the second eigenvalue of theLaplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every n-vertex graph of genus g and maximum degree d satisfies lambda₂(G) = O((g+1)³d/n).This recovers the O(d/n) bound of Spielman and Teng for planar graphs, and compares to Kelner\´s bound of O((g+1)poly(d)/n), but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that lambda₂(G) = O(dh⁶log h/n) whenever G is K_h-minor free. This shows, in particular, that spectral partitioning can be used to recover O(radicn)-sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on lambda₂ for graphs which exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find small separators in a general class of geometric graphs. Moreover, while the standard "sweep\´\´ algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the result of Alon, Seymour, and Thomas that every excluded-minor family of graphs has O(radicn)-node balanced separators.