Optimizing quadratic forms of adjacency matrices of trees and related eigenvalue problems

Let A be an adjacency matrix of a tree T with n vertices. Conditions are determined for the existence of a fixed permutation matrix P that maximizes the quadratic form xtP tAPx over all nonnegative vectors x with entries arranged in nondecreasing order. This quadratic form problem is completely solved, and its answer leads to a corresponding solution for the problem of determining conditions for the existence of a fixed permutation matrix P that maximizes the largest eigenvalue of matrices of the form PDP t + A, over all real diagonal matrices D with nondecreasing diagonal entries. It is shown that there is a tree with six vertices for which neither of the problems has a solution, and all other trees with six or fewer vertices have solutions for both problems. By duality, the results also apply to the analogous problem of minimizing the smallest eigenvalue of matrices of the form PDP t + A. © 2001 Elsevier Science Inc. All rights reserved.