Numerical ranges of cyclic shift matrices

We study the numerical range of an n x n cyclic shift matrix, which can be viewed as the adjacency matrix of a directed cycle with n weighted arcs. In particular, we consider the change in the numerical range if the weights are rearranged or perturbed. In addition to obtaining some general results on the problem, a permutation of the given weights is identified such that the corresponding matrix yields the largest numerical range (in terms of set inclusion), for n <= 6. We conjecture that the maximizing pattern extends to general n x n cyclic shift matrices. For n <= 5, we also determine permutations such that the corresponding cyclic shift matrix yields the smallest numerical range. (c) 2023 Elsevier Inc. All rights reserved.