MINIMUM SPECTRAL-RADIUS OF A WEIGHTED GRAPH

M. Fiedler recently introduced the following question. What is the optimal distribution of nonnegative weights (with total sum one) among the edges of a given graph, so that the spectral radius of the resulting adjacency matrix is minimum? He himself has shown that the optimum solution is achieved by some decomposition of the given graph G into a collection of mutually vetex disjoint odd cycles and balanced bipartite subgraphs which maximizes a certain objective function. We present a polynomial time algorithm which finds this decomposition. Our approach is related to matching theory.