Path Resistance Method for Bounding the Smallest Nontrivial Eigenvalue of aLaplacian

摘要

We introduce the path resistance method for lower bounds on the smallestnontrivial eigenvalue of the Laplacian matrix of a graph. The method is based on viewing the graph in terms of electrical circuits: it uses clique embeddings to produce lower bonds on lambda2 and star embeddings to produce lower bounds on the smallest Rayleight quotient when there is a zero Dirichlet boundary condition. The method assigns priorities to the paths in the embedding: we show that for an unweighted tree T, using uniform priorities for a clique embedding produce a lower bound on lambda2 that is off by at most an O(log diameter(T)) factor. We show that the best bounds this method can produce for clique embeddings are the same as for a related method that uses clique embeddings and edge lengths to produce bounds.